IC-NR 


*B    272 


TAYLOR 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


Class 

\ ; 


SURVEYOR'S 
HAND    BOOK 


BY 


T.    U.    TAYLOR 

C.  E,,  University  of  Virginia,  M.  C.  E.,  Cotnell  University;  Professor  cf 

Civil  Engineering  in  the  University  of  Texas;  Member 

American  Society  of  Civil  Engineers. 


CHICAGO    AND   NEW   YORK 

THE  MYRON  C.  CLARK  PUBLISHING  CO. 
1908 


Copyright  1908 

By 
The  Myron  C.  Clark  Publishing  Co. 


DEDICATION 


TO    GEORGE    W.    BRACKENRIDGE 

SAN  ANTONIO,  TEXAS. 

A  Civil  Engineer  and  Patriot 

Who  for  a  Quarter  of  a  Century  Has  Been  an  Active  and  Useful 
Friend  of  Higher  Education  in  Texas. 


179777 


PREFACE. 

It  has  been  my  object  to  prepare  a  book  for  the  use  of  the 
surveyor  in  the  field,  of  convenient  size  and  scope,  and  one  that 
contains  all  the  essentials  for  ordinary  surveying.  It  is  too  much 
to  hope  that  there  are  no  errors  in  these  pages,  in  theory  or 
example.  The  preliminary  proof  has  been  examined  by  twelve 
experienced  surveyors  and  I  am  indebted  to  them  for  many 
valuable  suggestions. 

I  am  under  lasting  obligations  to  my  colleagues,  R.  A. 
Thompson,  Expert  Engineer  of  the  Texas  Railway  Commission; 
Edward  C.  H.  Bantel,  Adjunct-Professor  of  Civil  Engineering; 
and  Stanley  P.  Finch,  Instructor  in  Civil  Engineering  of  the 
University  of  Texas.  In  addition  to  this  assistance  I  have  been 
aided  by  valuable  suggestions  from  F.  Lavis  and  Halbert  P. 
Gillette,  and  from  the  following  leading  surveyors  and  engi- 
neers of  Texas :  C.  F.  H.  von  Blucher,  Gustav  Schleicher,  B. 
F.  Love,  and  W.  D.  Twichell. 

The  thanks  of  the  author  are  hereby  expressed  to  W.  &  L.  E. 
Gurley,  Keuffel  &  Esser,  Eugene  Dietzgen  Co.,  and  A.  Wissler 
for  many  illustrations  of  instruments. 

The  traverse  table  has  been  omitted,  as  the  ordinary  ones  are 
useless  for  angles  not  multiples  of  quarter  degrees,  and  the  large 
ones  are  books  in  themselves.  As  lands  become  more  valuable, 
the  transit  survey  is  demanded  where  angles  are  read  to  the 
nearest  minute,  and  for  such  surveys  the  small  traverse  tables  are 
of  no  avail. 

Tables  I,  II,  III  and  IV  are  taken  by  permission  from  Henck's 
"Field  Book,"  while  Table  V  is  from  Searles'  "Field  Engineering." 

T.  U.  TAYLOR. 

Austin,  Texas,  September  1,  1908. 


CONTENTS. 

CHAPTER  I. 
CHAIN  SURVEYING. 

Page 

1.  Gunter's  Chain , 1 

2.  Engineer's     Chain 2 

3.  Vara  Chain 3 

4.  Steel  Tapes 3 

5.  Standardized    Tapes 4 

6.  Metallic  Tapes 5 

7.  Pins 5 

8.  Range   Poles 6 

0.     Plumb-bob   , 0 

10.  Chaining 7 

11.  Chaining  Over  Hills  or  Across  Valleys 0 

12.  Chain    Survey 9 

13.  Chain  Problems 10 

14.  Correction  for  Temperature 13 

15.  Stretch  of  Tape  Due  to  Pull. 13 

16.  Correction  for  Sag ......  14 

17.  Erroneous  Lengths : 15 

18.  Erroneous  Areas 16 

19.  Linear  Units 17 

20.  Units  of  Land  Measure 18 

21.  Area  of  a  Triangle , 19 

22.  The  57.3  Rule 20 

23.  Applications  of  the  57.3  Rule 20 

24.  Pacing  Survey 21 

25.  Location  of  Houses 22 

26.  Survey  of  Farm  by  Chain  or  Pace 23 

CHAPTER  II. 
COMPASS  SURVEYING. 

27.  The  Bearing  of  a  Line 25 

28.  Azimuth 25 

29.  The  Compass 25 

30.  Reading  the  Bearing , . 27 

31.  How  to  Use  the  Compass 27 

32.  The  Vernier .-.',.  28 

vii 


viii  CONTENTS. 

Page 

33.     Declination  of  the  Needle 20 

'34.     Compass    Vernier 30 

35.  To  Set  Off  Declination 31 

36.  Changes  in  Declination 31 

37.  Result  of  Changes 32 

38.  Old    Lines 33 

39.  Magnetic  Bearing 33 

407  To  Find  the  Declination  for  Any  Special  Farm. .  .  34 

41.  Local   Attraction 3;, 

42.  Witnessing  a  Line  or  Corner 35 

43.  Typical  Field  Notes 36 

44.  Compass  Adjustments • 36 

CHAPTER  III. 

TRANSIT  SURVEYING. 

45.  The    Transit 38 

46.  Compass  Attachment 40 

47.  Vertical  Circle ' .  '40 

48.  Shifting   Center '  41 

49.  The  Reticule '.'.'.'.'.'.  '41 

50.  Setting  Up  the  Transit ,41 

51.  Motions 42 

52.  Use  of  the  Transit 42 

53.  The  Transit  as  a  Compass 43 

54.  Transit  Surveying ' '  .  43 

55.  Transit  Vernier 43 

56.  Example 44 

57.  Reference  Lines .44 

58.  Repeating    Method ..  45 

59.  To  Adjust  the  Plate  Levels 46 

60.  Line  of   Sight  Adjustment 46 

61.  Peg   Adjustment 48 

62.  Location  of  Meridian  by  Polaris .,.. ,  48 

63.  Circumpolar    Stars 51 

64.  Location  of  Meridian 52 

65.  PZS  Triangle '.'.'.'.'.'.'.'.'.'.  52 

66.  Formulas    53 

67.  Observation  on  Sun 54 

68.  Refraction    55 

69.  Solar  Attachment .55 

70.  Meridian  Without  Calculation 57 

71.  Example  58 

72.  Example  '59 


CONTEXTS.  ix 

CHAPTER  IV. 
CALCULATION  OF  AREAS. 

Page 

73.  Latitude  and  Departure  of  a  Course 61 

74.  Traverse    Tables 62 

75.  Example     63 

76.  Error  of   Closure 63 

77.  Balancing  a  Survey 64 

78.  The  Double  Meridian  Distance 66 

79.  Area  of  a  Farm 67 

80.  Area   Table 68 

81.  Courses  of  No  Latitude  or  Departure 72 

82.  Example  73 

83.  Area  by  Co-ordinates 75 

84.  Traversing 76 

85.  Example 76 

8f>.     Approximate    Traversing 77 

87.  Irregular  Boundaries 78 

88.  Discrepancies    79 

CHAPTER  V. 
DIVISION  OF  LAND. 

89.  Division  of  Triangle 81 

90.  Division  Line  Through  Internal  Point 82 

91.  Division  of  Quadrilateral 83 

92.  General  Solution 84 

93.  Case  1 85 

94.  Case  II. . -. 86 

95.  Case  III 87 

96.  Case  IV 89 

97.  Example  89 

98.  Dividing  Land 90 

99.  Example  90 

100.  Example  92 

CHAPTER  VI. 
LEVELING. 

101.  The  Y  Level 96 

102.  The   Telescope 96 

103.  Setting  Up  the  Instrument 100 

104.  Rods    100 

105.  Theory  of  Leveling 102 

106.  Bench    Marks 104 


CONTENTS. 

Page 

107.  Profiles   106 

108.  Crosswire    Adjustment 106 

109.  Bubble-Tube  Adjustment 106 

110.  Adjustment  of  Wyes 107 

111.  The  Radius  of  the  Bubble-Tube 107 

112.  Curvature  of  Earth 109 

113.  Vertical  Curves 110 

114.  Curve  in  Sag Ill 

115.  Vertical  Circular  Curves 112 

CHAPTER  VII. 
TOPOGRAPHIC  SURVEY. 

116.  Topographic    Survey 1 14 

117.  Topographic   Methods 114 

118.  Stadia   Formulas 115 

119.  Wire  Interval 1 1(> 

120.  Inclined    Sights 117 

121.  Stadia   Rod 118 

122.  Field  Work Ill) 

123.  Reduction    Methods 122 

124.  Colby's  Slide  Rule ....'. 123 

125.  Usual    Approximations 124 

126.  Topography  by  Hand  Level 120 

CHAPTER  VIII. 
RAILROAD    SURVEYING. 

127.  Railroad    Surveying 129 

128.  Degree    Formula 129 

129.  General  Formula 12!) 

130.  To  Lay  Out  Curve 130 

.131.  Obstacles    " 132 

132.  Location  by  Offsets 132 

133.  Middle  Ordinate 133 

134.  Approximate  Formulas " 133 

135.  Reduction    Tables .134 

136.  Metric    Curves 135 

137.  Preliminary   Survey 136 

138.  Location    Survey '. 136 

139.  Field   Book 138 

140.  Transit    Party 140 

141.  Stakes 141 

142.  Hubs   141 

143.  Hand-Level   . 142 


COXTEXTS.  xi 

Page 

144.  Slope  Stakes  in  Excavation 144 

145.  Slope  Stakes  in  Embankment 147 

146.  Berms 149 

CHAPTER  IX. 
EARTHWORK. 

147.  Prismoidal    Formula , 151 

148.  Railroad    Excavation ; 152 

149.  Level   Sections 153 

150.  Two-Level   Sections 153 

151.  Three-Level  Sections .......  154 

152.  Irregular  Sections , .  154 

153.  Rules   , i55 

154.  Side  Hill  Cuts •.... 156 

155.  Average  End  Areas 157 

156.  Error  of  Average-End  Area  Formula 158 

157.  Examples 159 

158.  Preliminary  Estimates 160 

159.  Earthwork"  Note-Book 161 

160.  Special  Case 163 

161.  Borrow  Pits .164 

162.  End  of  Fill 166 

163.  Overhaul .166 

164.  Shrinkage 167 

CHAPTER  X. 
CITY  SURVEYING. 

165.  The  City  Engineer . 171 

166.  Objects   of   Survey .171 

167.  Monuments 172 

168.  Additions 173 

169.  Kinds  of  Monuments 173 

170.  Location  of  Monuments 175 

171 .  Tapes 175 

172.  Transit '. 176 

i  73.  Datum 178 

174.  General    Maps .....179 

175.  Water-Pipe  Map 179 

176:  City  Blocks 180   , 

1 77.  Rectangular    Blocks ..,  180 

178.  Rectangular    Lots 181 

179.  Irregular  Blocks  and  Lots 182 

180.  Private  Notes , 183 


xii  CONTENTS. 

Page 

181.  Prescriptive    Rights , ....184 

182.  Cross- Section   of    Streets 184 

183.  City  Engineering   Records 185 

184.  Field   Note-Books ,, . 18(i 

185.  Detail    Maps 187 

186.  Orders,  Bids.  Etc 189 

CHAPTER  XI. 
PLOTTING  AND  LETTERING. 

187.  Plots    190 

188.  Protractor  Method. 190 

189.  Latitude  and  Departure  Method 191 

190.  The  Tangent  Method 191 

191.  The  Sine  Method .'191 

192.  Co-ordinate    Method 192 

193.  Correcting  the  Plot 194 

194.  Lettering    194 

CHAPTER  XII. 
GOVERNMENT  SURVEYING. 

195.  Radii  of  Parallels. 190 

196.  Angular  Convergence  of  Meridians 196 

197.  Linear  Convergence 197 

198.  Off-Sets    , 198 

199.  Running  Parallels 198 

200.  Tangent   Method 199 

201.  Secant    Method 199 

202.  Intermediate    Off-Sets 199 

203.  Example  200 

204.  Reference  Meridians  and  Standard  Parallels 200 

205.  Ranges    202 

206.  Townships   202 

207.  Dividing  Up  a  Township 203 

CHAPTER  XIII. 
TRIGONOMETRIC  FORMULAS. 

208.  Formulas   for  Right  Triangle 208 

209.  Solutions  for  Right  Triangle 208 

210.  Oblique  Triangle 209 

211.  Right  Spherical  Triangle 210 

212.  Oblique  Spherical  Triangle 210 


CONTEXTS,  xiii 

CHAPTER  XIV, 

TABLES, 

Page 

Table  I.     Logarithms    of    Numbers 211 

Table  II.     Logarithmic    Sines,    Cosines,    Tangents    and 

Cotangents    227 

Table  III.     Natural   Sines  and  Cosines 275 

Table  IV     Natural  Tangents   and  Cotangents. 285 

Table  V.     Cubic   yards    per    100   feet    for    slopes    %:1, 
%:1,  1:1,  1%:1,  2:1,3:1,.., ,  ....;..„..„.  .299 


CHAPTER  I. 
CHAIN  SURVEYING. 

1.  Gunter's  Chain.  —  This  chain  was  invented  in  1620  by 
Edmund  Gunter.  an  English  surveyor,  and  is  now  in  use  in  a 
majority  of  the  older  states  of  America.  Previous  to  its  inven- 
tion, chains  of  irregular  lengths  had  been  in  use,  but  there  was 
no  uniform  system,  and  as  soon  as  Gunter's  chain  was  invented 
it  was  generally  adopted. 

This  chain  is  66  ft.  or  792  ins.  in  length,  and  is  divided  into 
100  equal  parts,  called  links,  each  link  being  7.92  ins.  long. 
Eighty  of  these  chains  make  one  mile.  Also,  we  know 

1  acre  =  4,840  sq.  yds.  =  43,560  sq.  ft. 

1  sq.  chain  =  66  X  66  =  4,356  sq.  ft. 

43.560 
'  .  1  acre  -=    r--  sq.  chains,  =  ••  10  sq   chains. 


Distances  are  measured  in  full  chains  and  decimals.  If  the 
distance  between  two  points  is  9  full  lengths  and  S3  links,  we 
call  the  length  9.83  chains,  as  each  link  is  one-hundredth  part  of 
a  chain.  Each  link  is  composed  of  three  parts  —  a  long  wire  with 
looped  ends  and  two  rings.  These  rings  can  be  left  open  or 
soldered  (brazed).  If  left  open  they  soon  become  oval  and 
elongated  in  the  direction  of  the  chain,  and  the  chain  thus  be- 
comes lengthened.  It  is  therefore  best  to  have  all  joints  brazed, 
as  this  makes  the  ring  connections  more  stable  and  less  liable  to 
stretch.  Figure  1  is  an  illustration  of  one  form  of  Gunter's 
chain  and  the  two  rings  at  each  joint  can  be  seen  in  the  upper 
right-hand  part  of  the  figure.  At  each  end  of  the  chain  there 
are  two  brass  handles,  the  measuring  length  cf  the  chain  being 
from  back  to  back  of  the  handles.  These  loop  handles  are  at- 
tached to  the  chain  by  nuts  that  are  intended  to  be  adjustable. 
When  fixed  in  one  position,  it  is  important  that  they  remain 
stationary  till  adjusted  by  the  surveyor.  The  wearing  surfaces 
make  it  necessary  to  remove  links  and  this  renders  the  chain  in- 

1 


2  SURVEYOR'S    HAND    BOOK. 

accurate  for  fractional  parts  of  a  chain.  There  are  in  all  about 
600  wearing  surfaces  and  if  each  surface  is  worn  one-hundredth 
part  of  an  inch,  the  chain  will  be  lengthened  6  ins.  and  this 
would  produce  an  error  of  1  per  cent  in  the  calculation  of  areas. 
The  tenth  link  from  the  end  is  marked  by  a  one-point  brass 
tag,  the  twentieth  by  a  two-point  tar,  the  thirtieth  by  a  three- 
point,  the  fortieth  by  a  four-point,  and  the  fiftieth  by  a  round  tag, 
it  being  the  middle  of  the  chain.  At  the  center  there  is  generally 
a  snap  link  for  disconnecting  the  chain,  so  that  a  half-chain  can 


Fig.    1. 

be  used  for  steep  hills  and  rough  country.  The  chain  is  folded 
by  commencing  at  the  middle  and  folding  two  links  at  each  time 
in  the  form  of  a  warped  surface,  making  when  completed  a 
shape  something  like  an  hour-glass.  In  unfolding  the  chain,  take 
both  handles  in  one  hand  and  with  the  other  throw  the  chain 
from  you.  With  a  little  practice  this  can  be  done  so  that  it  will 
stretch  its  full  half-length  when  thrown  and  the  whole  chain 
can  then  be  opened  out. 

2.     Engineer's  Chain. — This  chain  is  made  similar  to  Gun- 
ter's  chain,  is  100  ft.  in  length  from  back  to  back  of  the  handles, 


CHAIN    SURVEYING  3 

and  is  tagged  every  10  ft.  Each  link  is  1  ft.  long  and  it  con- 
sists of  one  long  wire  and  two  or  three  rings  whose  joints  are 
brazed.  This  chain  is  now  rarely  used  in  railroad  or  city  sur- 
veying where  great  accuracy  is  required.  The  steel  tape  has  al- 
most wholly  superseded  it  where  accurate  work  is  desired. 

3.  Vara  Chain. — The  vara  chain  is  20  varas  long,  and 
each  vara  is  divided  into  five  equal  parts.  Each  vara  is  marked 
by  a  tag  with  its  distance  from  one  end  stamped,  and  the  tags 
are  numbered  from  1  to  19.  The  chain  is  thus  divided  into  100 
equal  parts,  each  part  being  one-fifth  of  a  vara  or  one  one- 
hundredth  of  20  varas,  and  is,  therefore,  0.2  of  a  vara.  It  is 
necessary  to  remember  this,  for  in  the  Gunter  and  engineer 
chains  the  chain  itself  is  the  unit  of  length.  If  the  distance  be- 
tween two  points  is  five  full  lengths,  16  varas  and  2  links,  then 


Fig.    2. 

the  distance  is  116.4  varas.     In  Texas  a  vara  is  33^  ins.  long  by 
law. 

4.  Steel  Tapes. — For  precise  measurements  the  steel  tape 
(Fig.  2)  is  used.  It  varies  in  length  from  3  ft.  to  1,000  ft.,  and  is 
made  of  the  best  steel  reasonably  flexible.  The  tape  has  the  ad- 
vantage of  having  no  wearing  surfaces,  and  is  easily  folded  or 
looped  up  like  a  rope.  The  width  of  the  tape  varies  from  3/16 
to  Vz  in.,  the  thickness  being  about  1/64  in  It  is  marked  every 
5  ft.  from  one  end  and  numbered  on  brass  and  copper  plates 
bent  around  the  tape  from  5  to  95  ft.,  and  every  foot  is  marked 
by  a  brass  rivet,  and  each  foot  from  the  end  is  divided  into 
tenths  of  a  foot.  The  even  5-ft.  marks  are  usually  made  on  a 
brass  plate  or  sleeve,  and  the  even  10-ft.  marks  are  made  on  a 
copper  sleeve.  In  order  to  assist  in  identifying  the  even  10-ft. 


4  SURVEYOR'S    HAND    BOOK. 

marks  when  the  figures  have  become  so  worn  that  they  are 
illegible,  rivets  are  driven  through  the  plate  close  to  the  sleeve, 
one  at  the  10  and  90-ft.  marks,  two  at  the  20  and  80- ft.  marks, 
three  at  the  30  and  70-ft.  marks,  and  four  at  the  40  and  60-ft. 
marks.  The  rivets  are  always  driven  between  the  sleeve  and  the 
50-ft.  mark,  so  that,  by  noticing  the  position  of  the  rivets,  it  is 
easy  to  distinguish  the  proper  point.  The  50-ft.  mark  is  marked 
by  two  rivets,  one  on  each  side  of  the  sleeve. 

One  of  the  best  fcrms  of  steel  tapes  for  railroad  or  city  engi- 
neers and  surveyors  is  about  1A  in.  wide  and  has  the  numbers  for 
the  different  foot-marks  stamped  on  solder  which  adheres  to  the 
tape.  This  form  of  tape  has  the  advantage  of  not  having  a 
shoulder  or  projection  to  catch  against  the  reel  when  the  tape  is 
being  wound  up  or  run  out,  or  to  catch  on  stones  or  other  rough 
objects  while  in  use. 


Fig.    3. 

5.  Standardized  Tapes. — For  accurate  base  line  measure- 
ments a  steel  tape  from  100  to  300  ft.  long  is  used  (Fig.  3). 
Such  tapes  should  be  standardized;  that  is,  the  absolute 
length  between  the  marked  points  under  a  certain  pull  at  a 
known  temperature  should  be  determined.  This  is  generally  done  in 
this  country  by  the  United  States  Coast  and  Geodetic  Survey 
(Washington,  D.  C.)  for  a  nominal  price.  If  it  is  necessary  to 
use  any  tape  unsupported,  its  correct  length  when  hanging  free 
may  be  found  by  direct  comparison.  Lay  the  tape  on  a  smooth 
straight  support,  give  it  the  proper  pull  and  mark  the  end  points  ; 
then,  holding  one  of  the  ends  directly  over  one  of  the  marks  just 
made,  give  it  a  known  pull.  Drop  a  plumb  line  from  the  other 
end  of  the  tape  and  notice  the  amount  by  which  it  differs  from 
the  second  mark.  In  this  way  the  correct  length  of  the  unsup- 
ported tape  under  any  given  pull  may  be  determined. 


CHAIN    SURVEYING. 


5 


6.  Metallic  Tapes. — The  most  serviceable  tape  for  ordi- 
nary or  common  use  is  the  metallic  tape  (Fig.  4),  which  is  a 
cloth  tape  manufactured  with  very  fine  brass  wires  interwoven 
into  it.  This  tape  is  generally  %  in.  wide,  and  is  made  in 
lengths  of  25,  50  ttnd  100  ft.  It  is  conveniently  inclosed  in  a 
leather  case,  and  when  it  is  rolled  up  it  can  easily  be  carried  in 
the  pocket.  For  light  and  irregular  work  it  is  much  more  con- 
venient than  the  larger  steel  tapes.  It  is  largely  used  in  building 
construction,  cross-section  work,  and  in  railroad  engineering,  and 
in  many  places  where  its  lightness,  compactness,  and  flexibility 
commend  it.  It  can  not  be  used  where  accuracy  is  very  im- 


Fig.    4. 

portant,  for  it  stretches  considerably  under  pull,  but  after  a 
short  period  of  use  it  will  be  found  to  have  become  permanently 
stretched. 

7.  Pins. — Surveying  pins  are  used  to  keep  tally  of  the 
number  of  chains  measured.  They  are  made  of  pieces  of  round 
steel  wire  3/16  in.  in  diameter  and  about  16  ins.  long.  One  end 
is  pointed  and  the  other  is  bent  to  form  a  ring  or  handle 
about  2  ins.  in  diameter,  Fig.  5.  Eleven  such  pins  form  a 
set,  and  they  are  carried  on  a  key  ring,  about  6  ins.  in  diameter, 
made  of  the  same  sized  wire.  Each  pin  usually  has  a  small  piece 


SURVEYOR'S    HAND    BOOK. 


of  red  flannel  tied  to  its  handle  so  as  to  make  it  more  easy  to  be 
found  when  used  in  the  field.  In  railroad  chaining  stakes  are 
generally  used  instead  of  pins  and  these  stakes  are  driven  at 
every  full  station  (every  100  ft.)  and  at  intermediate  points  be- 
tween the  stations.  For  a  description  of  stakes  see  the  chapter 
on  Railroad  Surveying. 

8.  Range  Poles. — Range  poles  are  rods  of  steel,  or  wood- 
en rods  shod  in  steel,  varying  from  6  to  10  ft.  in  length.  Alter- 
nate foot  lengths  of  the  rod  are  painted 
red  and  white  to  make  it  more  readily 
distinguishable  against  any  background. 
They  are  used  by  the  rear  chainman  to 
keep  in  a  straight  line  when  chaining.  If 
the  sun  is  shining  and  long  sights  are 
taken,  the  bright  part  of  the  range  pole 
is  seen  as  the  other  part  is  in  shadow. 
To  avoid  this,  a  range  pole  with  a  flat 
face  is  used  with  the  central  longitudi- 
nal line  clearly  denned  and  alternate 
foot-lengths  on  each  side  of  this  line 
painted  black  or  red. 

9.  Plumb-Bob. — In  chaining  over 
rough  or  inclined  ground  it  is  often 
necessary  to  raise  one  end  of  the  chain 
or  tape  to  bring  it  to  a  horizontal.  To 
locate  a  point  on  the  ground  directly  un- 
der such  elevated  points  or  ends  o-f 
chain,  a  plumb-bob  will  have  to  be  used. 
The  usual  form  of  a  plumb-bob  is  shown 
in  Fig.  G,  (a)  and  (b),  which  consists  of  a  conical  shaped  body 
rounded  into  a  neck  and  head  in  the  upper  part.  The  bottom  or 
apex  of  the  cone  is  usually  tipped  with  a  steel  point,  while  the 
cap-screw  at  upper  end  has  a  hole  through  its  center  for  the  in- 
sertion of  the  cord  by  which  it  is  suspended.  In  the  ordinary 
form  (a)  the  cap  screw  is  taken  off,  the  cord  is  inserted,  and  a 
knot  tied  in  the  cord  to  prevent  its  slipping  through  the  cap- 
screw  when  the  bob  is  suspended.  Fig.  6  (b)  is  a  special  form 


Fig. 


CHAIN    SURy  EYING, 


of  plumb-bob  which  is  provided  with  a  spool  on  the  inside  by 
which  the  cord  can  be  wound  up  and  carried  inside  the  bob  in- 
stead of  being  wrapped  around  the  outside  as  in  the  ordinary 
form.  This  winding  is  done  by  turning  the  cap-screw  at  the  top. 
10.  Chaining. — A  line  is  measured  or  chained  by  two 
men,  called  the  rear  and  head  chainman.  They  should  start  with 
eleven  pins,  the  rear  chainman  taking  one  pin  and  holding  his 
end  of  the  chain  or  tape  over  the  intial  point,  and  lines  or 
ranges  the  head  chainman  in  with  the  distant  flag.  The  head 


Fig.    6. 

chainman  sticks  a  pin  at  this  point  and  advances  to  another  sta- 
tion, the  rear  chainman  following  to  the  station  just  left  by  him. 
The  rear  chainman  places  his  end  of  the  chain  or  tape  over  this 
station,  and  again  ranges  in  the  head  chainman.  The  rear  chain- 
man must  be  careful  to  collect  all  pins,  ^nd  when  the  head  chain- 
man calls  "Out"  he  must  drop  his  end  of  the  chain  and  go  to  the 
head  chainman,  and  should  hand  him  10  pins.  The  head  chain- 
man should  count  the  pins,  and  if  there  are  not  10  pins  the  line 


8  SURVEYOR'S    HAND    BOOK. 

should  be  chained  over.  The  number  of  "Outs"  is  recorded  by 
each  chainman.  If  we  are  usinj  the  surveyor's  chain,  and  have 
three  "outs,"  and  the  head  chainman  has  measured  3  chains  and 
23  links  on  the  new  out,  the  lenfth  of  the  line  is  33.23  chains. 
The  head  chainman  always  starts  on  a  new  out  with  10  pins,  and 
the  head  chainman  or  the  rear  chainman  should  never  have  more 
than  10  pins  in  his  hand  while  measuring.  The  initial  point  and 
the  end  of  each  out  should  be  carefully  marked  so  that  if  a  mis- 
take is  made  in  a  long  line  the  chammen  can  return  to  the  last 
out,  and  not  have  to  go  back  to  the  beginning  of  the  line.  The 
methods  of  keeping  track  of  the  "outs"  vary  with  different  sur- 
veyors. In  chaining  long  lines,  a  string  tied  in  the  button  hole 
of  the  coat  or  shirt,  with  segments  of  unequal  length,  can  be 
used  by  tying  a  knot  in  the  long  segment  for  an  ordinary  "out," 
and  one  in  the  short  segment  for  every  ten  "outs."  Another 
method  is  to  have  the  chainmen  make  tally  marks  in  a  note  book. 

In  chaining  up  a  hill  which  is  too  steep  for  one  length  to  be 
brought  horizontal,  the  head  chainman  stretches  the  chain  to  its 
full  length,  and  then  returns  and  takes  a  point  on  the  chain  suffi- 
ciently near  the  rear  chainman  to  pull  that  part  horizontal.  He 
marks  the  point  on  the  ground  under  the  selected  point  with  his 
pumb-bob,  or  places  the  point  on  the  chain  immediately  on  the 
ground,  the  rear  chainman  drops  his  end  of  the  chain  and  takes 
up  the  point  selected  by  the  head  chainman  and  raises  it  as  high 
as  he  can  over  the  point  as  tested  by  his  plumb-bob.  The  head 
chainman  in  the  meantime  selects  another  point  qn  the  ground 
in  advance  and  marks  that  on  the  ground  as  before.  This  process 
is  repeated  until  the  length  of  the  chain  is  exhausted.  This  is 
called  "breaking  the  chain."  In  "breaking  the  chain"  it  is  well 
to  take  sections  of  the  chain  that  are  multiples  of  ten. 

In  measuring  down  a  hill,  the  process  is  reversed,  so  that  the 
rear  chainman  holds  his  end  on  the  ground  or  near  it,  and  the 
head  chainman  holds  his  point  over  his  head  as  high  as  he  can. 

The  chain  or  tape  should  always  be  held  level,  because  the 
horizontal  distance  between  the  two  fixed  points  is  constant,  not- 
withstanding the  fact  that  changes  may  be  made  on  the  surface 
of  the  ground.  In  the  early  days  surveyors  paid  no  attention  to 


CHAIN    SURVEYING. 


holding  their  chain  level,  and  there  has  resulted,  in  consequence 
many  discrepancies  in  their  surveys,  and  much  litigation.  All 
good  surveyors  are  now  very  careful  in  observing  this  rule.  In 
using  the  tape  in  rough  countries  or  thick  underbrush,  it  is  a 
good  plan  where  great  accuracy  is  demanded  to  attach  the  handle 
of  the  tape  by  a  short  loop  of  strong  cord  to  allow  twisting  of 
the  tape  without  breaking. 

11.  Chaining  Over    Hills  or  Across  Val- 
leys.— When  it  is  impossible  to  see  one  sta-     L/ 
tion  from  the  initial  station  on  account  of 

an  intervening  hill  or  high  timber,  a  series 
of  range  poles  is  used  and  a  random  line 
marked  out  so  that  at  least  three  points  can 
be  seen  from  one  station. 

Given  the  two  points  A  and  B  (Fig.  7),  to 
set  the  range  poles  in  line  AB.  We  start  out 
from  A  and,  guessing  at  the  line,  set  enough 
range  poles  in  a  random  line  AD  so  that  at 
least  two  can  be  seen  from  B.  Then  the  man 
at  B  will  have  the  flag  pole  at  3  set  over  in 
the  line  B-2  to  the  point  4,  the  man  at  4  will 
have  the  Hag  2  set  over  in  the  line  4-1  at  5, 
the  man  at  o  will  have  the  flag  1  set  over  in 
the  line  5- A.  Then  again  flag  4  will  be  set 
over  to  some  point  'nearer  AB,  in  line  between 
B  and  5,  etc.  This  process  is  repeated  until  all 
the  range  poles  are  in  the  line  A-B.  In  the 
preliminary  ranging  in  the  men  themselves  can 
act  as  range  poles.  Only  one  man  is  abso-  p^g  7 

lutcly  necessary  if  he  has  plenty  of  range 
poles,  but  two  can  do  it  with  reaspnable  efficiency. 

12.  Chain  Survey. — When   the  area   of  a  farm  is   wanted, 
or  if  it  is  desired  to  construct  a  map  of  same,  it  may  be  divided 
by    stations   into   a   system  of  triangles.     All  the  sides   are  then 
measured  carefully  and   a.  map   of   the  triangulation   system   can 
then  be  made  to  scale.     The  buildings  and  other  topographical 
features,  such  as  roads,  fences,  etc.,  can  be  tied  in  by  measuring 


10 


SURVEYOR'S    HAND-   BOOK. 


AB  =  240 


from   the   nearest  stations   and  a  sufficient  number  of  points  on 
the  building,  and  map  can  be  completed  to  scale. 

In  a  recent  survey  (Fig.  8)  the  following  measurements  were 
made  : 

EA  =  180 

££  =  300 

In  case  it  were  impossible  to  measure 
the  line  EB,  the  area  may  still  be  found  by 
a  chain  survey  if  use  is  made  of  two  aux- 
iliary lines  AF  and  BF,  the  point  F  being 
in  AE  produced.  By  means  of  these  aux- 
iliary lines  the  triangle  BAF  may  be  calcu- 
lated and  hence  the  angle  BAE  becomes 
known.  From  this  angle  and  the  sides  AB 
and  AE  the  length  of  BE  can  be  calcu- 
lated and  the  area  found  as  before. 

PROBLEM  1. — Make  a  map  of  the  chain 
survey  ABCDE  to  a  scale  of  1  in.  =  50 
units. 

13.     Chain  Problems. —  (a)    To  erect  a 
perpendicular  to  a  line  at  any  point : 

We  know  that  if  the  sides  of  a  tri-        D 

In  K 

angle  are  3,  4  and  5,  or  any  multiple  of 
these,  the  triangle  will  be  right.  This 
is  apparent,  as  the  sum  of  the  squares  of 
3  and  4  equals  the  square  of  5.  If, 
in  the  triangle  AKB,  Fig.  9,  the  sides 
are  18,  24  and  30,  it  will  be  a  right  tri- 
angle. 

The  rear  chainman  holds  his  end  of 
the  chain  at  B  in  the  line  BK  so  that 
the  distance  BK  is  equal  to  18  links;  he 
also  holds  the  end  of  the  seventy-second 
link  at  the  same  point ;  the  head  chain- 
man passes  the  chain  around  a  pin  at  K, 
which  has  been  firmly  driven  or  pushed  &{%  9 


30 


CHAIN    SURVEYING. 


11 


into  the  ground,  then  takes  hold  of  the  forty-second  link  and 
stretches  the  chain  so  that  all  parts  are  taut.  A  pin  is  then 
driven  at  A,  which  determines  the  perpendicular  AK. 

In    reality   there    are    a   great    number   of   ways   in   which   the 
problem  can  be  solved,  for"  if 

2w  =  first  side 
M2  —  1  =  second   side 
w2  +  1  =  third   side 

the  triangle  is  a  right  triangle,  as  (2n)2  +  (n2  —  I)2  =  4w2  +  w4  — 
2/r  +  l  =  „*  -f  2/r  +  1  =  (V  4-  I)2. 

Therefore,  we  can  make  n  equal  to  any  number  greater  than 
unity.      The  following  are  some  of  the  numbers  actually  used: 


3, 

4, 

5, 

5, 

12, 

13, 

6, 

8, 

10, 

8, 

15, 

17, 

7, 

24, 

25, 

12, 

35, 

37, 

11, 

60, 

61,  etc. 

Fig-    10. 

(b)  Another  easy  method  of  erecting  a  perpendicular  to 
the  line  AK  at  the  poinj:  K  (Fig  10)  is  to  let  one  of  the  chain- 
men  hold  the  end  of  the  chain  at  K,  while  a  second  chainman 
holds  the  other  end  of  the  chain  at  any  point  on  AK  so  that  the 
chain  will  be  slack.  The  middle  point  of  the  chain  is  then  car- 
ried away  from  the  line  AK  until  it  occupies  the  position  AEK. 
Tf  the  end  of  the  chain  at  K  is  now  swung  around  until  it 
reaches  a  point  C  in  the  same  straight  line  as  A  and  E,  the  line 
CK  will  be  the  perpendicular  to  AK  at  K. 


12 


SURVEYOR'S    HAND    BOOK. 


(c)  To  find  the  distance  across  a  marsh,  river  or  pond  by 
use  of  the  chain : 

Suppose  a  line  that  we  are  chaining  reaches  a  point  A,  Fig. 
11,  and  a  river  intervenes  wider  than  the  length  of  one  chain, 
and  we  wish  to  find  the  distance  AB.  At  the  point  A  by  the 
former  method  we  measure  the  distance  to  K  on  the  perpendicu- 
lar AK,  and  at  the  ponit  K  set  off  the  right  angle  BKC ' ,  and 
mark  where  KC  produced  crosses  our  original  line.  Measure 
AC. 


Fig.    11. 

In  the  right  triangle  BKC 

AK2  = 


Fig.    12. 


X  AC 


AK* 
BA-'-   -JO- 

Caution.  —  AK  should  be  taken  at  least  one-half  of  AB,  oth- 
erwise AC  will  be  so  short  that  a  slight  error  in  measuring  will 
produce  a  large  error  in  the  result. 

(d)  Similar  Triangles:  To  find  the  distance  AB,  Fig.  12, 
erect  a  perpendicular  to  AB  at  B  with  a  chain  and  prolong  it 
to  some  point  C  ;  measure  BC  and  set  a  flag  pole  at  D  in  the 
line  DC.  Erect  a  perpendicular,  DE,  to  BD  and  have  the  flag- 
man move  along  this  perpendicular  until  he  is  in  the  line  ACE. 


CHAIN    SURVEYING.  13 

Set  the  flag  pole  firmly  in  the  ground  and  measure  DC  and  DE. 
AB  :  DE  : :  BC  :  DC 

DE   X   BC 
AB= DC~ 

14.  Correction  for  Temperature. — Steel  tapes  are  stand- 
ardized by  the  Coast  Survey  by  comparison  with  known  stand- 
ards at  Washington,  and  each  standardized  tape  is  marked  some- 
what as  follows :      ''Length  100  feet  at  temperature  62°  F.,  pull 
12  pounds  horizontal." 

The  average  coefficient  of   linear  expansion  is  0.0000065  for 
each  degree  Fahrenheit,  and  each  unit  length. 
Let  L  =:  Length  of  tape. 
C  —  Coefficient  of  tape. 
7"— Rise  in  temperature. 

Then  the  increased  length  of  the  tape  — LCT. 
Total    length   of   tape  =  L  +  LCT  =  L(1  +  CT~). 
EXAMPLES.     A  300-ft.  tape   was  standardized  at  62°  F.,  pull 
12  Ibs.      A  base  line  was  measured  when  the  temperature  of  the 
tape  was  102°  F.,  find  the  length  of  the  tape. 
Increase  =  LCT. 

—  .0000065  X  300  X  40  =  .078. 
Total  length  =  300  +  .078  •=  300.078  ft. 

15.  Stretch  of  Tape  Due  to  Pull. — It  is  necessary  to  sub- 
ject all  tapes  to  what  is  called  a  standardized  pull  for  their  true 
lengths.     If  it  takes  a  12-lb.  pull  to  make  a  tape  100  ft.  long,  any 
pull  greater   than  this  will  stretch  the.  tape,  and  has  to  be  al- 
lowed for. 

Let  P  —  pull  in  pounds. 

A=  cross-section  in  square  inches. 

P  _ 
Then  the  pull  per  unit  area  —  *j  —  unit  stress. 

If  S  =  total  stretch 
L  =  length  of  tape 

Unit  stretch  =  - 


14  SURVEYOR'S    HAND    BOOK. 

In  ordinary  pulls  the  unit  stretch  varies  directly  as  the  unit 
pull. 

Unit  pull          PL 
Therefore,  Unit  stretch  =  AS  =  Constant- 

This  is  Hook's  law,  which  was  published  in  the  form  "ut 
tcnsio  sic  vis."  The  unit  pull  divided  by  the  unit  stress  is  con- 
stant within  the  elastic  limit  and  is  called  "the  coefficient  of 
elasticity,''  and  is  gcneraly  represented  by  the  letter  E.  For 
steel  E  =  30,000,000  Ibs. 

EXAMPLE:     A  bar  I%"x%"x20"  long  was  subjected  to  a  pull 
of  18,000  Ibs.  and  produced  a  stretch  of  %  in.     Find  E. 
Area  =  9/8  sq.  in. 

18000 
Unit  pull  =  -T-  =  16,000  Ib. 


Unit  stretch  =  240  = 

£  =  liioon  divided  by  1/1920=  10000x1020  =  30,720,000. 
EXAMPLE:  If  a  100-  ft.  tape  was  standardized  at  a  pull  of 
12  Ibs.,  and  has  a  cross-section  of  .00371  sq.  in.,  find  how  much 
it  will  be  stretched  by  a  pull  of  26  Ibs.  if  £  =  30,000,000.  The 
stretch  over  the  standard  length  will  be  due  to  the  extra  pull 
of  14  Ibs. 

S  =  total  stretch  in  feet. 

Unit  stretch  =  ^QQ 

Unit  pull  -  Iwn 

1400 


30,000,000  = 


30,000,000  X  .00371 

16.  Correction  for  Sag.  —  The  foregoing  corrections  for 
pull  and  temperature  assume  that  the  tape  is  horizontal,  but  hi 
field  measurements  it  is  never  horizontal,  although  the  two 
ends  may  be  in  the  same  horizontal  plane.  The  tape  hangs  in 
a  curve,  which  is  practically  a  parabola,  with  which  a  circular 


CHAIN    SURVEYING.  15 

curve  can  coincide  almost  exactly.      The  effect  is  to  shorten  the 
chain. 

-  If  rf=sag 

L==  length  of  tape  or  chain 

8d2 
The  correction  for  sag  =  ^ 

EXAMPLE  :  A  100-ft.  tape,  standardized  at  62°  F.  and  12  Ibs.  pull 
was  used  to  measure  a  line  when  the  temperature  was  92°  F.,  pull 
25  Ibs.  and  sag  0.5  ft.  Find  the  correct  length  of  the  tape  if  the 
cross-section  is  0.00,3  sq.  in. 

Correction   for   temperature  =  .0000065  X  100  X  30  =  0.0195 

Correction  for  sag=  8/300  X  0.25  =  0.0067 

13  X  100  X    1,000 
Correction  for  pull          3  X  30,000,000 

Length  of  tape  =  300  +  0195  —  .0067  +  .1444  =  300.1572 

PROBLEM  2. — A  100-ft.  tape,  cross-section  1/300  sq.  in.,  was 
standardized  at  62°  F.  and  pull  12  Ibs.  Find  the  length  for  a 
temperature  of  96°  F.,  pull  28  Ibs.  and  a  sag  of  0.5  ft. 

PROBLEM  3.— A  standardized  tape  is  100  ft.  long  between 
marks  at  61°  F.,  and  a  pull  of  11  Ibs.  Find  the  length  when 
temperature  is  97°  F.,  20  Ibs.  pull,  and  a  sag  of  0.70  ft.,  if  cross- 
section  is  1/300  sq.  in. 

17.  Erroneous  Lengths. — Chains  become  changed  by  the 
breaking  of  links,  the  loss  of  handles,  and  the  wearing  of  the  600 
rubbing  surfaces.  In  the  use  of  the  chain  two  points  on  the 
ground,  66  ft.  or  100  ft.  apart,  should  be  marked,  and  the 
chain  should  be  compared  with  this  at  frequent  intervals.  The 
outer  edge  of  one  of  the  handles  is  placed  over  the  zero  and 
the  100-ft.  mark  is  marked  by  a  file  if  the  chain  is  too  long.  If 
distances  are  measured  by  chains  that  are  too  long,  we  can  find 
the  true  lengths  of  the  lines  by  calculation  without  measure- 
ment. If  the  length  of  the  chain  used  is  100  +  a,  and  in  the 
measurement  we  called  it  100  ft,  then  the  length  of  the  line 'as 
measured  will  be  too  short. 

If  the  extra  length  of  the  chain  is  due  to  wear  or  stretch 
throughout  the  length,  the  true  length  of  a  line  that  has  been 


16  SURVEYOR'S    I1AXD    BOOK. 

measured  with  a  tape  of  erroneous  length  may  be  found  by 
multiplying  the  true  length  of  the  tape  by  the  number  of  times 
it  was  applied  to  the  ground  in  measuring  the  line.  Afti-r  a 
line  9.864  chains  in  length  had  been  measured  it  was  found  that 
the  chain  was  really  100.25  ft.  long,  find  the  true  length  of 
the  line.  The  chain  was  applied  to  the  line  9.804  times,  con- 
sequently its  true  length  must  be  9.8G4  X  100.25  =  988.866  ft. 

However,  it  might  happen  that  one  link  of  an  engineer's  chain 
had  been  broken  and  t^ken  out.  thus  making  the  chain  99  ft. 
long.  Suppose  an  engineer's  chain  was  used  in  measuring  a 
line  the  length  of  which  was  recorded  as  628  ft.,  and  it  \vuk- 
then  discovered  that  1  link  was  out  of  the  10-ft.  section  next  to 
the  head  chainman.  What  is  the  true  length  of  the  line?  Six 
full  lengths  were  measured  =  6  X  99  =  594  ft.  If  the  28  ft.  was 
measured  with  the  end  of  the  chain  next  to  the  rear  chainman 
the  true  length  of  the  line  was  622  ft.,  but  if  the  28  ft.  was 
measured  with  the  part  of  the  chain  that  contained  the  unknown 
missing  link,  then  the  true  length  of  the  line  was  621  ft. 
Let  a  =  assumed  length  of  chain, 

/=true  length  of  chain, 
M  =  measured   length  of  line  as  measured  with  chain  of 

erroneous  length, 
r  =  true  length  of  line, 

ft  =  number  of  chain  lengths  in  M  (whole  or  fractional.) 
Then,  A/  =  na 
T  =  nt. 


18.     Erroneous  Areas.  —  If  a  farm  is  surveyed  with  a  chain 
of  erroneous  length  and  the  area  is  calculated  by  use  of  the  er 
roncous  data,  we  can  find  the  area  without  rechaining. 
Let  C  =  calculated  area  of  farm, 
X  =  true  area  of  farm, 
<r=  assumed  length  of  chain, 
/  =  true  length  of  chain, 

Then,  na=-  measured  length  of  side  of  farm, 
»/  =  true  length  of  same  farm. 


CHAIX    SURVEYING.  17 

Now,   similar   polygons   are  to  each  other   as  the  square   of 
their  homolgous  ^ides. 

.'.  A'  :C  ::  (ntY'  :  («fl)2. 


It  is  well  to  observe  that  in  the  re-calculations  for  correct  length 
of  a  line  or  for  correct  area  of  a  farm  the  assumed  length  al- 
ways appears  as  denominator  of  correction  ratio.  This  assumed 
length  is  usually  an  even  number,  and  is  generally  20,  66,  100,  etc. 
EXAMPLE:  A  line  was  measured  with  a  chain  that  was  sup- 
posed to  be  100  ft.  long;  the  length  of  the  line  as  measured  was 
iMJ.4  ft.  In  testing  the  chain  immediately  afterwards  it  was 
found  to  be  100.25  ft.  long.  Assuming  that  the  stretch  was  pro- 
portional throughout,  find  the  length  of  the  line. 

1  no  ?^ 
Correct  length  of  line  =  986.4  X  =988.866 


PROBLEM  4.  —  The  assumed  length  of  a  chain  is  100  ft.,  the 
calculated  area  99.01  acres.  The  true  length  of  the  chain  was 
found  after  the  calculation  to  be  99  ft.  6  ins.  Find  the  true 
area. 

PROBLEM  5.  —  A  chain  used  to  measure  a  field  was  .100  ft.  2  ins. 
in  length,  and  it  was  assumed  in  measuring  the  farm  to  be  100 
ft.  long.  If  the  calculated  area,  based  on  the  erroneous  length 
of  chain,  was  11.72  acres,  find  the  true  area. 

PROBLEM  6.  —  A  farm  was  surveyed  with  Gunter's  chain  and 
the  area  was  calculated  to  be  39.6  acres.  The  chain  was  tested 
immediately  after  the  survey  was  made,  and  it  was  found  to  be 
4  ins.  too  long.  Find  the  true  area  of  the  farm. 

PROBLEM  7.  —  If  the  calculated  area  was  133%  acres  and  the 
vara  chain  was  used  in  chaining  which,  after  the  survey, 
was  found  to  be  3%  ins.  short,  find  the  true  area. 

19.  Linear  Units.  —  The  yard  is  the  primary  unit  of  length 
in  the  English  measure.  The  standard  yard  is  the  distance  be- 
tween two  points  at  a  certain  temperature  on  a  bar  of  platinum 
kept  in  London  in  the  office  of  the  Chancellor  of  Exchequer  of 
Great  Britain.  A  copy  of  this  is  kept  in  Washington,  D.  C 


18  SURVEYOR'S    HAND    BOOK. 

An  inch  is  one-thirty-sixth  part  of  a  yard,  and  a   foot  is  one- 
third  part  of  a  yard,  or  \'2  ins. 

To  convert  feet  to  varas  multiply  by  «>..".»;. 

To  convert  yards  to  varas  multiply  by  1.08. 

To  convert  Guntcr's  chains  to  varas  multily  by  23.76. 

To  convert  poles  to  varas  multiply  by  •"• !'!. 

To  convert  meters  to  varas  multily  by  1.1811. 

20.     Units  of  Land  Measure.— 

One  acre  =  4840  square  yards, 

=  43560  square  feet, 

—  10  square  chains. 

=  160  square  poK-. 

=  5645.376  square  varas, 

=  4046.87  square  meters. 
Onevara  =  33%  inches. 
One  yard  =  36  inches. 
One  foot  =  .36  vara. 

10,000 
One  square  vara=  1111.1  square  inches=       y       square  ins. 

One  square  yard  =  1296   square   inches. 

A  Spanish  league  was  denned  as  a  square,  5,<HH)  varas  on 
a  side. 

One  league  =  25,000,000  square  varas. 

=  4428.203  acres. 

One  labor  =  a  square  of  1000  varas, 
=  1,000,000  square  varas, 
=  177.128  acres, 
=  1/25  of  a  league.  ,    | 

One  linear  mile  =  1900.8.  varas. 
One  meter  =  39.37  inches. 
One  linear  mile  =  1609.35  meters. 

A  labor  was  assigned  by  the  Mexican  government  to  settlers 
for  the  purposes  of  agriculture,  hence  the  name;  while  a  league 
was  assigned  for  grazing  purposes.  In  this  way  a  league  and 
labor  became  associated. 


CHAIN    SURVEYING 


19 


21.  Area  of  a  Triangle.  —  By  geometry  we  know  that  the 
area  of  a  triangle  =  Vz  (p  X  c)  =K,  where  />  represents  the  al 
titude  CD  and  c  the  base  of  any  triangle. 


Fig.    13. 

In  the  right  triangle  ADC,  p*=tr—x*. 
In  the  right  triangle  BDC,  f>*=a2—(c— 


*= 


(^  +  r2  —  a2)2 
=  (2fcf  +  &2+cs  —  a2)  (2bc  +  a2  —  b2  —  c 

=  [(fr  +  O2  —  fl8)  x  (o1—  (&  —  Of.l  ^4 

=  (&  +  c  +  a)  (&  +  r  —  a)  (a  —  &  +  O  ( 

Let  2J=(a  +  &  +  r) 

Then    (b  +  c  —  a)=2s  —  2a  =2  (s  —  a 


=2s  —  2c  =  2(s  —  r) 


•'    ^ 


45    5-a)(5- 


— a)  (5—6)  (5— c) 


Therefore  i(/v)  =  K"  =  V5(5— a) (5— 6) (5— c) (S) 

PROBLEM  8.— Calculate  the  areas  of  triangles  ABE,  EEC,  and 

EDC  in  Fig.  8. 

PROBLEM  9.— If  the  sides  of  a  triangle  are  520,  560,  and  600 

varas,  find  the  area  in  acres. 


20  SURVEYOR'S    HAND    BOOK. 

PROBLEM  10. — If  the  sides  of  a  triangle  are  13,  20  and  21 
chains  (66  ft),  find  the  area  in  acres. 

PROBLEM  II. — If  a  —  750  varas,  b  =  (>">0  varas,  c  =  *200  varas, 
find  area  in  acres. 

PROBLEM  12. — If  a  =  50  poles.  b  =  4l  poles,  c  =  39  poles,  find 
area  in  acres. 

PROBLEM  13.— If  a  =  300  poles,  6  =  240  poles,  c  =  180  poles, 
find  area  in  acres. 

PROBLEM  14.— If  a=  280  poles,  fc  =  224  poles,  c=168  poles, 
find  area  in  acres. 

22.  The  57.3  Rule.— Let  EOA  (Fig.  14)  be  a  triangle 
where  the  angle  x  is  less  than  6°,  and  the  two  arms  OA  and  OE 
practically  equal.  If  with  O  as  a  center  and  OA  as  a  radius  we 
describe  a  circle  passing  through  E  we  have: 


Fig.    14. 

X°  :360°  ::y  :  2irr 
where  y  =  AE 

360       v        57.3y 
Then  X°=  -^   X  '-  =  — ~    (4) 

That  is,  the  small  angle  in  degrees  times  the  long  side  is 
equal  to  the  short  side  times  57.3. 

PROBLEM  15.— A  straight  roadway  1,320  ft.  long  has  a  rise  of 
21  ft.  above  the  horizontal  through  the  low  end.  Find  its  angle 
of  elevation. 

23.  Applications  of  the  57.3  Rule. — If  the  angle  AOE 
Fig.  14,  equals  one-tenth  of  57.3°,  then  we  have 

5°.73  AOB=    '  Distance      ' "'  Distance  =  10  X  offset 
That  is,  when  the  small  angle  is  5°.73  or  5°  44',  the  -distance  is 
ten  times  the  small  side  or  offset. 


CHAIN    SURVEYING.  21 

If  the  angle  EOA  is  equal  to  0°.573,  that  is,  34'.38,  the  long 
side  is  one  hundred  times  the  offset.  Hence  OA  =  100  X  AE. 

This  is  generally  expressed  by  saying  that  the  distance  is  100 
times  the  offset.  This  principle  is  used  in  finding  the  approxi- 
mate area  of  a  boundary.  The  angle  that  OA  makes  with  some 
reference  line  is  measured,  and  the  distance  OA  is  found  by 
making  the  angle  equal  to  34.38  minutes.  The  assistant  at  A 
attaches  one  end  of  a  tape  or  chain  to  the  point  A  and  then 
takes  AE  at  right  angles  to  AO  and  is  sighted  in  the  line  OE 
by  the  distant  transitman.  When  he  is  located,  he  reads  on  the 
tape  the  distance  AE  and  records  it  in  his  note  book.  The  dis- 
tance from  A  to  the  instrument  man  is  100  times  this  distance 
AE. 

PROBLEM  16. — Make  a  drawing  of  the  following  area  to  a 
scale  of  1  in.  equals  100  ft.,  and  find  the  area  in  acres,  by  divid- 
ing the  boundary  into  triangles. 

Point.  Angle.  Offset. 

A   0°  8.50  feet 

B   - 45°  10.00 

C :  75°  9.40 

D  90°  9.60 

E  120°  8.60 

F  ' 150°  7.20 

G  180°  6.00 

24.  Pacing  Survey. — A  rough  approximate  idea  of  the 
area  of  a  farm  can  be  .obtained  by  a  pacing  survey.  With  a  little 
practice  a  man  can  train  himself  to  step  off  a  yard  at  each  stride 
and  in  this  way  a  fair  approximation  can  be  made  to  the  area  of 
a  small  farm  or  parcel  of  land.  In  a  farm,  ABCDF,  Fig.  15,  let 
AB  350  yds. ;  BC  400  yds. ;  CD  90  yds. ;  DE  266.3  yds. ;  EF  250 
yds. ;  FA  281.8  yds.  Now  the  area  of  the  farm  can  be  found  by 
dividing  the  field  into  triangles  or  by  locating  the  points.  CDE. 
etc..  by  offsets  from  some  reference  line,  AB.  If  the  land  is 
divided  into  triangles  we  pace  the  distance  BD  410  yds.,  BE  300 
and  AE  211.  This  divides  the  land  into  four  triangles,  BCD, 
BDE.  BEA  and  AEF.  The  area  can  be  calcnbted  by  the  use  of 
formula  (3). 

If  it  is  desired  to  locate  the  corners  by  offsets,  we  adopt  some 


22 


SURVEYOR'S    HAND    BOOK. 


reference  line  from  which  to  take  offsets.  This  reference  line 
need  not  be  a  side  of  the  farm,  but  can  be  some  line  assumed  for 
convenience.  However,  in  the  case  of  Fig.  15,  we  shall  assume 
AB  as  the  reference  line.  As  ABC  is  a  right  angle,  the  distance 
EC  400  yds.,  will  locate  C,  and  as  angle  BCD  is  also  right,  the 
distance,  CD  90  yds.,  will  determine  the  point  D.  Let  DG  be  a 
perpendicular  from  D  on  line  AB.  If  a  perpendicular  be  dropped 
from  E  on  AB  cutting  AB  at  H,  where  £H  =  240<and  HE  =  180, 
the  point  E  is  determined.  The  point  F  is  similarly  located  where 
perpendicular,  MF  250  and  BM  480.  The  areas  of  the  trapezoids, 

BCDG,  GDEH,  HEFM,  and 
that  of  the  triangle,  AMF,  can 
be  found  to  be  respectively 
36,000,  43,500, 51,600,  and  16,250 
square  yards  and  the  area  of 
ABCDEF  114,850  sq.  yds. 

Instead  of  trying  to  regulate 
the  stride  to  1  yd.,  some  prefer 


Fig.   15. 


to  take  the  usual  stride  used  in 
walking,  counting  the  number 
of  steps  it  takes  to  cover  100 
ft.  and  then  estimate  the  dis- 
tance. Thus  if  it  takes  40  steps 
for  100  ft.  and  there  are  114 
steps  in  the  length  of  the  line, 
the  number  of  feet  is  found  by 
multiplying  114  by  100  and  dividing  by  40.  In  this  case  the  line 
would  be  about  280  ft.,  or  93  yds. 

25.  Location  of  Houses. — These  can  be  located  by 
range  lines,  regular  offsets,  or  by  intersections. 

Range  Lines. — Let  MN,  Fig.  16,  be  the  base  line  or  line  near- 
est any  given  corner  of  the  house,  FDCE.  Have  a  range  pole 
set  at  A  in  line  MN  and  in  range  with  CD,  the  side  of  the  build- 
ing; and  another  set  at  B  in  line  MN  and  in  range  with  CE,  an- 
other side  of  the  house.  Pace  the  distances  from  A  and  B  to 
end  of  base  line  MN.  On  the  map  locate  AB  on  line  MN  and 
with  AB  as  a  diameter  draw  a  circle.  With  5  as  a  center  and  a 


CHAIN    SURVEYING. 


23 


radius  BC ;  cut  circle  at  C.  Join  BC  and  AC  and  produce  BC 
in  EC  and  AC  in  CD.  Lay  oft  CE  and  CD  to  scale  and  locate 
the  rest  of  the  building. 

Rectangular  Offsets.— Let  CH  and  EK,  Fig.  16.  be  the  per- 
pendiculars from  corners  of  house  on  base  line  MN.     Pace  EK, 


fi 


B 

17. 


Fig. 
The  house  can  thus  be 


Fig.    16. 

CH,  and  KH  to  end  of  base  line  MN. 
located  on  the  map. 

By  Intersection. — Let  MN,  Fig.  17,  le  base  line  and  A  and  B 
two  points  in  this  line.  Pace  distances  AB,  BC,  BE,  AE,  CD, 
EC,  and  tTie  distance  from  A  or  B  to  end  of  base  line.  To  locate 
house  on  map,  locate  A  and  B  on  map  and  with  A  as  center  and 
AE  as  radius  draw  arc  and  with  B  as  cen- 
ter and  BE  as  radius  draw  arc  cutting  first 
arc  at  E.  This  locates  E.  Then  with  E  as 
center  and  CE  as  radius,  draw  an  arc  and 
with  B  as  center  and  BC  as  radius  draw 
arc  cutting  the  other  arc  at  C.  Draw  CD 
perpendicular  to  EC  and  lay  off  CD  to 
scale,  and  through  D  and  E  draw  DP 
and  EF  parallel  to  CE  and  CD  respec- 
tively. 

26.  Survey  of  Farm  by  Chain  or  Pace. —  The  exact 
area  of  a  farm  ABCDEF,  Fig.  18,  can  be  found  by  use  of  the 
chain  or  tape,  or  an  approximate  estimate  of  the  area  can  be 
found  by  pacing  the  sides  and  diagonals.  In  Fig.  18  the  follow- 
ing lengths  of  sides  were  found :  AB  =  170  yds.,  BC  =  492 
yds.,  CD  =  296  yds,,  DE  =  272  yds.,  EF  =  286  yds,  FA  =  260 


18. 


24  SURVEYOR'S    HAND    BOOK. 

The  configuration  of  the  ground  was  such  that  the  farm 
could  be  divided  into  triangles  by  running  diagonals  from  the 
corner  A.  These  diagonals,  AC,  AD,  and  AE,  were  found  to  be 
488,  436,  and  322  yds.,  respectively.  The  area  of  ABC  was 
found  by  formula -(3)  to  be  41,000  sq.  yds.;  that  of  ACD,  63,760; 
that  of  ADE,  43,680;  and  that  of  AEF,  35,376,  making  a  total 
area  of  the  whole  farm  of  183,816  sq.  yds.  =  37.98  acres. 

To  check  the  foregoing  calculation  a  point  P  was  taken  on  a 
knoll  near  the  center  of  the  farm  and  the  following  distances 
were  paced :  PA  =  196,  PB  =  312,  PC  =  350,  PD  =  240,  PE 
=  206,  PF  =  337.  The  areas  calculated  by  formula  (3)  nn> 
PAB  =  14,873,  PBC  =  54.577.  PCD  =  34.538,  PDE  =  23.546, 
PEF  =  33,504,  PFA  =  25,422,  or  a  total  of  184,460  Sq.  yds.  = 
38.1  acres.  If  the  distances  are  all  carefully  chained  instead  of 
paced  the  two  methods  should  check  within  one-tenth  of  an  acre. 


CHAPTER  II. 
COMPASS  SURVEYING. 

27.  The  Bearing  of  a  Line. — The  acute  angle  that  a  line 
makes  with  the  meridian  is  called  its  true  bearing.     If  the  acute 
angle  is  made  with  that  part  of  the  meridian  to  the  north  of  us 
it  is  called  north,  and  if  in  addition  it  cuts  to  the  right  it  is  called 
North  X°  East,  where  X  equals  the  acute  angle.     If  the  acute 
angle  is  made  with  that  part  of  the  meridian  to  the  south  of  us 
and  cuts  to  the  right  it  is  called  SX°W.     In  Fig.  19  the  bearing 
of  AB  is  N  32°  E;  that  of  AD,  N  54°  W;  that  of  EF,  S  61°  W, 
while  that  of  EG  is  5"  27°  E. 

28.  Azimuth. — The  azimuth  of  a  line 
is  the  angle  made  with  the  true  meridian, 
and  is  measured  from  the  south  around  by 
the  west,  north,  and  cast  to  the  south  again. 
If  the  bearing  of  a  line  is  5  30°  W ',  the  azi- 
muth is  .°,n°  ;  if  the  bearing  is  N  39°  W,  the 
.-izimuth  is  141°;  if  the  bearing  is  .V  39°  E, 
the  n.ximuth  is  219°,  and  if  the  bearing  is  S1 
39°  E,  the  azimuth  is  321°.     In  some  states 
it  is  the  practice  to  define  "bearing"  as  the 
acute  angle  made  by  a  line  with  the  mag- 
netic meridian  (that  is,  with  the  needle  in 
its  mean  position). 

29.  The   Compass. — The    essential   parts    of   a    surveyor's 
compass  (Fig.  20)  are  a  magnetic  needle,  a  graduated  horizontal 
circle,  and  a  line  of  sights.    These  conditions  can  be  fulfilled  very 
crudely  or  elaborately.     It  is  also  convenient  to  have  a  declina- 
tion arc  attached  to  the  compass  on  which  we  can  set  off  the 
declination  of  the  needle.     A  magnetic  needle  when  poised  freely 
will  not  point  towards  the  North  Pole,  but  will  dip  towards  the 
north  an  amount  of  x  degrees.     To   make  it  horizontal   in  the 
compass  it  is  mounted  on  an  agate  pivot  and  the  South  end  is 
weighted  by  having  an  adjustable  brass  wire  at  that  end.     The 

25 


Fig. 


SURVEYOR'S    HAND    BOOK. 


accuracy  of  the  compass  depends  largely  upon  the  activity  of  the 
needle,  which  depends  upon  the  intensity  of  the  magnetic  force, 
which  must  be  kept  alive.  The  pivot  upon  which  the  needle  is 


mounted  is  in  the  center  of  a  grad- 
uated circle  which  is  generally  raised 
to  the  level  of  the  ends  of  the  needle 
and  is  graduated  on  a  silver  plate. 
Inside  the  compass  hox  we  find  tin- 
letters  £,  5",  W,  and  AT.  If  the  com- 
pass has  no  declination  arcs  the 
zeros  are  in  the  line  of  sights  as  de- 
termined by  the  slots  in  the  stand- 
ards or  uprights.  The  graduated  cir- 
cle* is  mounted  on  a  brass  plan 
which  has  extended  arms,  to  which 
the  uprights  are  attached  by  me  -in- 
of- mill-head  screws.  If  the  arms  are 
not  extended  the  uprights  are  at- 
tached to  the  graduated  circle  and 
fold  down  over  the  face  when  not  in 
u-e.  To  set  off  the  declination  ac- 
curately, each  compass  should  be  pro- 
vided with  a  declination  arc  with  a 
vernier  attached. 

For  the  purpose  of  leveling,  the 
compass  is  provided  with  two  bubble 
tubes  whose  axes  are  at  right  angles 
to  each  other.  It  is  leveled  by  a 


Fig.   20. 

ball      and      socket      joint      which      affords      easy      and      quick 
methods    of    setting    up.      It    can    be    mounted    on    a     Jacob's 


COMPASS    SURVEYING.  27 

staff  or  a  tripod,  but  in  most  cases  county  surveyors  use  the 
Jacob's  staff  on  account  of  its  ease  of  transportation.  The  ball 
and  socket  joint  is  mounted  on  the  Jacob's  staff,  which  has  a 
sharp  conical  iron  shoe.  In  setting  up,  the  staff  is  driven  into 
the  ground  two  or  three  times  to  get  a  firm  footing  so  that  there 
will  be  no  vibration.  The  compass  is  then  set  on  the  staff — 
leveled — and  it  is  now  ready  for  use.  When  moving  from  sta- 
tion to  station  the  compass  should  always  be  removed  from  the 
staff  and  carried  under  the  arm,  with  the  needle  screwed  reason- 
ably tight.  In  setting  up  always  loosen  the  ball  and  socket  joint 
and  have  the  compass  almost  level  and  along  the  line  of  sights 
before  tightening.  If  the  tripod  is  used  the  compass  can  be 
taken  off  in  moving  from  one  station  to  another  or  it  can  be 
left  on  as  with  the  transit.  The  tripod  gives  much  more  ac- 
curate work  than  the  Jacob's  staff  because  you  can  locate  the 
points  more  accurately,  and  it  gives  a  much  more  stable  support. 

30.  Reading  the  Bearing. — To  read  the  bearing  of  a  line, 
set  up  the  instrument  over  any  point  on  the  line,  turn  the  com- 
pass so  that  the  arrow  in  the  compass  box  points  in  the  direc- 
tion in  which  you  are  running  the  line,  and  read  the  north  end 
of  the  needle.    The  north  end  of  the  needle  will  lie  between  two 
letters,  one  of  which  will  be  N  or  S,  while  the  other  will  be  E 
or  W .     If  it  lies  between  N  and  W ',  the  bearing  is  northwest ;  if 
between  ^  and  W ,  the  bearing  is  southwest,  etc.     In  sighting  al- 
ways place  the  eye  at, the  end  of  the  compass  box  marked  S. 

31.  How  to  Use  the  Compass. — Set  up  the  tripod  with 
the  legs  wide  apart  and  firmly  pressed  into  the  ground.     Place 
the  compass  on  the  brass  spindle  and  then  fasten  the  sights  by 
means   of  the  thumb  screws   provided   for   that   purpose.     This 
spindle  is  connected  with  the  head  of  the  tripod  by  a  ball  and 
socket  joint,  which  gives  it  a  limited  range  of  motion.    A  groove 
about  %  in.  wide  and  about  the  same  depth  is  cut  in  the  spindle, 
which   engages  a  pin   piercing  the   socket   of  the  compass    body 
which  fits  over  the  spindle  and  prevents  the  compass  from  fall- 
ing off  the  tripod.     Take  hold  of  the  compass  with  both  hands 
and  level  it  by  means  of  the  motion  available  in  the  ball-and- 
socket  joint.     When  both  bubbles  are  in  the  center  of  their  run — 


28 


SURVEYOR'S    HAND    BOOK. 


that  is,  in  the  center  of  the  tube — the  instrument  is  level.  Do 
not  lower  the  needle  until  the  compass  has  been  leveled.  The 
compass  may  now  be  pointed  in  any  direction  by  turning  it  on 
the  spindle  axis.  In  moving  the  instrument  to  another  point, 
raise  the  needle  by  means  of  the  screw  controlling  it,  remove 
the  compass  from  the  tripod  by  pulling  in  the  small  pin  in  the 
socket  mentioned  above,  at  the  same  time  lifting  the  compass 
from  the  tripod.  Carry  the  compass  under  one  arm  and  the 
tripod  in  the  other  hand,  or  on  the  other  shoulder.  If  a  Jacob 
staff  is  used  instead  of  the  tripod,  the  brass 
spindle  connected  to  the  ball-and-socket 
joint  is  connected  with  the  staff  by  a  tight 
fitting  joint.  When  the  compass  is  placed 
in  its  box  to  be  stored  away  the  needle 
should  be  left  free. 

During  some  seasons  of  the  year  the 
compass  will  be  affected  by  a  charge  of 
electricity  due  to  atmospheric  conditions. 
When  this  is  the  case  one  end  of  the  needle 
will  often  adhere  to  the  glass  plate.  If  the 
glass  is  touched  with  a  damp  substance  it 
will  relieve  this  condition  and  release  the 
needle. 

32.  The  Vernier. — The  vernier  is  an 
auxiliary  scale,  either  straight  or  circular, 
designed  to  read  to  a  certain  given  part  of 
the  finest  division  on  the  limb.  Thus  in  the 
Xew  York  rod  (Fig.  21)  the  smallest  division  that  can 
be  read  from  the  rod  itself  is  one  one-hundredth  of 
a  foot,  but  the  vernier  cuts  this  part  into  ten  parts, 
so  that  we  can  read  to  one  one-thousandth  of  a  foot. 
In  the  ordinary  transit  the  finest  division  is  a  half  degree,  but 
with  the  aid  of  the  vernier  we  can  read  to  minutes.  If  AB  is  the 
limb  and  CD  is  the  vernier  scale,  let  a  equal  the  length  of  each 
part  of  the  limb,  and  b  equal  the  length  of  each  part  on  the 
vernier,  and  n  equal' the  number  of  parts  on  the  vernier,  then 
(ft  —  1)  will  be  the  number  of  parts  on  the  limb,  so  arranged 


Fig.   21. 


UNIVERSITY! 


COMPASS    SURVEYING. 


that  n  parts  on  the  vernier  is  equal  to  n  —  1  parts  on  the  limb, 
consequently  nb  =  a  (n  —  1). 

If  the  lowest  mark  on  the  vernier  agrees  with  a  mark  on  the 
limb,  then  the  highest  point  on  the  vernier  will  agree  with  a 
mark  on  the  limb,  also  the  second  mark  on  the  vernier  will  not 
agree  by  an  amount  of  a  —  b.  If  the  vernier  is  moved 
a  distance  a  —  b,  then  mark  No.  1  on  the  vernier 
will  agree  with -a  mark  on  the  limb;  if  moved  twice  this  dis- 
tance, then  mark  No.  2  will  agree  with  a  mark  on  the  limb ;  if 
moved  three  times  this  distance,  then  No.  3  will  agree  with  a 
mark  on  the  limb.  If  rrark  No.  3  on  the  vernier  agrees  with  a 
mark  on  the  limb,  it  means  that  the  zero  at  the  vernier  is 
3  (a  —  6)  from  above  the  nearest  point  on  the  limb. 

But  bn  =  a(n  —  1) 

a(n-  i) 

o  = 

n 

a(n  —  1)        a 
a  -  6  =  a  -       ~^-     ~  - 

a  —  b  is  always  one  nth  of  the  finest  space  on  the  limb  and  it 
is  called  the  fineness  of  reading. 

If  w  =  10  parts  and  a  =  1/100  of  a  foot,  then  the  vernier  reads 
to  1/1,000  of  a  foot. 

If  w  =  30  and  a  — 30'  then  the  vernier  reads  to  minutes.  This 
is  the  case  in  the  ordinary  transit;  a  =  30'  or  Vz°  and  ;i  =  30, 
and  we  can  read  to  minutes. 

33.  Decimation  of  the  Needle. — The  magnetic  needle  at 
any  point  when  mounted  on  a  pivot  and  weighted  at  one  end  so 
that  it  will  rest  in  a  horizontal  position  will  make  an  angle  with 
the  true  meridian.  This  angle  is  called  the  declination  or  varia- 
tion of  the  needle.  In  Texas  the  magnetic  meridian  cuts  to  the 
right  of  the  true  meridian  passing  through  a  point,  and,  there- 
fore, the  declination  is  said  to  be  east.  In  Austin  the  magnetic 
meridian  makes  an  angle  at  the  present  date  of  about  8°  with 
the  true  meridian,  or  the  declination  of  the  needle  is  said  to  be 
8°  east. 

The  line  of  zero  declination  (called  the  agonic  line)  now 
passes  near  Charleston,  S.  C. ;  Asheville,  N.  C. ;  Knoxville,  Tenn. ; 


30  SURVEYOR'S    HAND    BOOK. 

Lima,  Ohio;  Battle  Creek,  Mich.;  and  passes  through  the  re- 
mote corner  of  northeastern  Indiana.  All  sections  east  of  the 
line  have  west  declinations,  while  all  sections  west  of  this  agonic 
line  have  east  declinations. 

The  United  States  Coast  and  Geodetic  Survey  determines 
the  magnetic  declination  at  various  points  in  each  State  at 
stated  intervals;  and  by  this  means  not  only  is  the  declination 
accurately  determined,  but  its  rate  of  change  can  be  determined 
by  a  comparison  of  the  declination  for  different  dates.  These 
results  are  placed  on  a  map  (called  the  Isogonic  Chart)  by  the 
Coast  Survey.  This  chart  is  issued  at  least  every  ten  years  and 
is  of  great  use  to  surveyors,  as  it  gives  the  declination  for  all 
parts  of  the  United  States  with  reasonable  accuracy.  It  can  be 
obtained  by  addressing  a  letter  to  the  Coast  and  Geodetic  Survey, 
Washington,  D.  C 

VERNIER 


34.  Compass  Vernier. — One  form  of  compass  vernier  is 
shown  in  Fig.  22.  This  is  the  usual  form  of  the  vernier  on  the 
surveyor's  compass.  The  vernier  is  divided  into  30  equal  parts 
anc  these  30  parts  cover  29  parts  on  the  "limb"  or  graduated 
circle.  The  smallest  division  on  the  limb  is  one-half  degree  or 
30  minutes  and  as  the  vernier  can  read  to  one-thirtieth  of  the 
smallest  division  on  the  limb,  we  can  read  to  one-thirtieth  of 
30  minutes,  or  to  1  minute. 

We  further  notice  that  the  vernier-zero  is  nearer  the  5th 
division  of  the  limb,  and  we  find  that  the  5th  division  of  the 
vernier  to  the  left  of  the  vernier-zero  is  opposite  or  coincides 
with  a  division  on  the  limb.  Hence  the  reading  for  the  frac- 


COMPASS     SURVEYING.  31' 

tional  part  is  five  minutes,  which  corresponds  to  this  5th  divi- 
sion of  the  vernier  that  is  opposite  a  divison  of  the  limb.  The 
whole  reading  should  be  2°  30'  plus  5'  or  2°  35'. 

If  the  zero  of  the  vernier  is,  as  in  Fig.  22,  nearer  the  last 
division  between  the  two  zeros  than  it  is  to  the  division  beyond 
the  vernier-zero,  the  fractional  part  is  read  on  the  left  half  of 
the  vernier.  There  are  15  divisions  in  this  left  half  .and  if  the 
fractional  reading  is  between  zero  and  15,  one  division  of  the 
left  half  of  the  vernier  will  coincide  with  one  division  on  the 
limb,  and  the  number  of  this  division  on  the  vernier  is  the  frac- 
tional reading.  Thus  if  the  5th  division  on  the  vernier  agrees, 
as  in  Fig.  22,  with  a  division  on  the  limb,  the  reading  is  5' ;  if  the 
9th  agrees,  the  fractional  reading  is  9',  etc.  However,  if  the 
vernier  reading  is  greater  than  15',  this  reading  is  obtained  from 
the  upper  part  of  that  half  of  the  vernier  that  covers  a  section 
of  the  limb  reading. 

35.  To  Set  Off  Declination.— This  will  be  illustrated  by 
an  example.     Suppose  that  the  declination  of  the  needle  is  8°  15' 
east.     This  means  that  if  the  needle  was  allowed  to  swing  fre-ely 
it  would  come  to  rest  in  a  line  not  pointing  to  the  true  .north, 
but   in    a   line   that   makes   8°    15'  on   the   east   side   of   the   true 
meridian,  or  in  a  line  whose  bearing  is  N.  8°  15'  E.     To  set  off 
the  declination,  level  the  instrument,  lower  the  needle  and  allow 
it  to   come  to   rest.     Turn   the  compass  until  the   line  of  sight, 
through   the  slots   in « the    standards,   coincides- in   direction  with 
the  needle.     Clamp  the   instrument   in   this   position.     Since  the 
neeriU  when  at  rest  points  N.  8°   15'  E.,  the  line  of  sight  must 
how  be  N.  8°  15'  E.,  or  make  an  angle  of  8°  15'  with  the  true 
meridian.  Then  with  the  instrument  clamped,  and  without  disturb- 
ing the  line  of  sight,  turn  the  graduated  circle  in  the  compass 
box  by  means  of  the  milled-head  screw  until  the  needle   reads 
N.  8°   15'  E.     The  vernier  scale  that  marks  the  declination  arc 
should  now  read  8°   1-V.     The  final  and  accurate  test  is  the  ver- 
nier arc  where  all  declinations  should  be  set  off. 

36.  Changes  in  Declination. — The  declination  of  all  points 
west  of  the  agonic  line  has  been   decreasing,  while  that  to  the 
east    of    the    agonic    line    has    been    increasing.      In    Texas    the 


SURVEYOR'S    HAND    BOOK. 


declination  has  been  decreasing  at  the  rate  of  about  three  minutes 
per' year  since  the  time  of  the  first  recorded  land  patents.  This 
steady  annual  change  goes  through  a  large  series  of  years  and 
probably  goes  through  a  cycle. 

In  addition  to  the  annual  change  there  is  a  daily  change.  In 
Texas  the  needle  at  about  6  p.  m.  is  in  its  normal  position ;  at  8 
a.  m.  the  north  end  of  the  needle  swings  to  the  east  about  two 
to  three  minutes,  and  about  1  p.  m.  it  swings  about  the  same 
amount  to  the  west  of  the  normal  position. 

37.  Result  of  Changes.— An  old  survey  was  run  in  1864 
with  the  correct  declination  of  10°  at  the  time  the  survey  was 
made,  and  a  surveyor  in  1904,  not  knowing  the  present  correct 
declination  (which  is  8°),  sets  his  com- 
pass on  the  old  declination.  The  bear- 
ing of  line  was  N.  42°  E. ;  that  is,  the 
line  made  42°  to  the  right  of  the  true 
meridian  and  32°  with  the  magnetic 
meridian.  When  the  correct  declination 
was  set  off  in  the  compass  and  the  ends 
of  the  needle  were  brought  to  the  zero 
marks  on  the  graduated  circle  under  the 
glass  top,  the  line  of  sights  pointed 
along  the  true  meridian.  But  since  the 
original  survey  was  made  the  declina- 
tion has  decreased  to  8°  and  the  mag- 
netic meridian  has  turned  2°  to  the  left 

of  its  position  in  18(54.  Now  if  the  surveyor  of  1904  had 
set  off  the  declination  of  8°  on  the  declination  arc 
and  brought  the  ends  of  the  needle  to  agree  with 
the  zero  marks  of  the  graduated  circle,  the  line  of  sights 
would  have  pointed  along  the  true  meridian.  But  instead  he  set 
off  a  declination  of  10°,  and  when  he  made  the  ends  of  the  needle 
agree  with  the  zero  marks  the  line  of  sights  marked  out  was  2° 
to  the  left  of  the  true  meridian.  Now  if  AB,  Fig.  23,  were  the 
original  line  that  made  42°  with  the  true  meridian  AN,  and  AM 
were  the  position  of  the  magnetic  meridian,  the  magnetic 
meridian  of  1004  will  occupy  the  position  AM',  two  degrees  to 


Fig.   23.. 
its    position 


COMPASS     SURVEYING.  33 

the  left  of  AM.  The  surveyor  set  off  10°  on  the  declination 
which  made  the  line  of  sights  point  to  a  position  AN',  which  he 
assumed  to  be  the  true  meridian,  and  from  this  he  set  off  the 
bearing  42°.  As  the  angle  NAB  is  42°,  then  the  42°  measured 
from  AN'  will  fall  to  the  left  of  AB  in  some  position  AB'.  In 
all  cases  in  that  section  west  of  the  agonic  line,  where  the  sur- 
veyor uses  a  declination  greater  than  the  correct  declination,  he, 
in  effect,  turns  the  assumed  true  meridian  from  which  he  locates 
bearings  to  the  left,  and  all  lines  thus  run  will  fall  to  the  left 
of  the  old  lines,  if  said  old  lines  were  surveyed  with  the  correct 
declination. 

34.  Old  Lines. — In  surveying  old  lands  the  great  object 
is  to  ascertain  a  declination  which,  used  with  the  bearings  as  ob- 
tained from  the  field  notes,  will  retrace  the  old  lines.  This  is  the 
prime  object.  It  may  be  the  correct  declination  for  the  time  and 
place,  and  it  may  not.  If  two  points  A  and  B  can  be  found  on 
any  side  of  the  tract,  set  the  compass  at  one  of  these  points  A, 
run  a  random  line  AC  with  an  assumed  declination  and  the 
bearing  of  the  line  a  distance  AC  equal  to  the  distance  AB. 
Measure  the  distance  BC ' ,  multiply  it  by  57.3  and  divide  the 
product  by  the  length  of  the  line  AB.  The  result  is  the  error  in 
degrees  in  the  assumed  declination.  If  the  declination  is  cor- 
rected by  this  error  the  old  bearings  will  trace  out  the  lines  as 
formerly  marked. 

39.  Magnetic  Bearing.— In  some  of  the  older  States  the 
bearing  of  a  line  is  defined  as  the  angle  it  makes  with  the  mag"- 
netic  meridian.  The  result  of  this  is  in  all  of  the  States  west  of 
the  agonic  line  where  the  declination  has  been  decreasing  for 
years  that  the  northeast  and  southwest  bearings  will  be  increased 
over  the  old  bearings  by  an  amount  equal  to  the  change  in  declina- 
tion, while  the  northwest  and  southeast  bearings  will  be  de- 
creased by  the  change  in  declination  since  the  old  line  was  sur- 
veyed. In  the  country  east  of  the  agonic  line  the  reverse  of  the 
above  is  true. 

PROBLEM  17. — If  the  bearing  of  a  line  with  reference  to  the 
magnetic  merdian  in  the  States  west  of  the  agonic  line  were  N. 
72°  18'  E.,  declination  8°  45'  east  at  the  time  of  the  old  survey, 


34  SURVEYOR'S    IIAXD    BOOK. 

find  its  magnetic  bearing  at  the  time  \vhen  its  declination  was  7° 
'IV  east.  Find  the  declination  if  the  magnetic  bearing  of  the 
line  were  S.  36°  21'  E. 

PROBLEM  18. — The  magnetic  bearing  of  a  line  in  a  State  east 
of  the  agonic  line  was,  when  the  original  grant  was  surveyed  in 
1806,  S.  C8°  E.,  with  a  declination  of  25'  west.  Find  the  magnetic 
bearing  in  1806  when  the  declination  had  increased  to  2°  05'. 
Find  the  magnetic  bearing  if  the  true  bearing  was  S.  29°  42'  W. 

PROBLEM  10. — Find  the  magnetic  bearing  in  the  following : 


A. 
B 

True 
Bearing. 
N  26°  54' 

N  74°  12'  W 

Declin-                    Magnetic 
ation.                      Bearing. 
7°  54'  E                     
7°  54'  E                    

r 

•  N  33°  28'  W 

7°  54'  E                    

o 

S   >;)60  36'  E 

7°  54'  E                    

F 

N    2°  14'  E 

8°  17'  E                    

F 

S   87°  14'  E 

8°  17'  E                    

G 

N     5°  20'  W 

8°  17'  E        '            

f{ 

N  88°  ->2'  W 

8°  17'  E                    

A 

PROBLEM  20.  —  Find  the  true 
"Magnetic 
Bearing. 
N     3°  14'  W 

bearing  for  the  following  courses  : 
DecliiiT                       True 
ation.                       Bearing. 
2°     8'  E                    

H 

S     5°  18'  W 

6°  12'  E 

C 

1\     8°  16'  W 

3°  16'  W                  

/) 

S  74°  °6'  W 

3°  18'  W 

F 

N  17°  23'  W 

3°  lk>'  E 

F 

S   74°  -V  \\ 

4°  02'  W                  

G 

N  17°  23'  W 

5°  43'  E                      

If 

N    9°-  25'  E 

8°  56'  E 

40.     To  Find  the  Declination  for  Any  Special  Farm. — To 

resurvey  an  old  farm  or  tract  of  land  obtain  the  field  notes  from 
the  county  clerk's'  office  or  from  the  deeds  or  grants.  These 
papers  should  give  the  declination  used  in  the  original  survey. 
This  former  declination  (whether  right  or  wrong)  can  not  be 
used  in  a  subsequent  survey,  and  it  is  the  surveyor's  first  duty  to 
ascertain  the  proper  declination  to  use  in  his  own  survey.  If  he 
can  find  one  side  of  the  tract  marked  by  corners  or  trees,  he  can 
use  these  as  a  basis.  If  two  corners  at  the  end  of  a  line  can  be 
found,  all  he  has  to  do  is  to  set  off  a  declination  on  the  declina- 


COMPASS     SURl  'EYIXG.  3~> 

tion  arc  that  will  cause  the  compass  when  set  on  the  line  with 
the  true  bearing,  to  coincide  with  the  line  as  defined  by  the 
trees  or  corners.  However,  if  the  corners  can  not  be  seen  from 
each  other,  the  surveyor  must  select  a  declination  that  he  thinks 
will  be  correct.  With  this  declination  he  runs  a  random  line 
with  the  old  bearing  the  full  length  of  the  line,  and  marks  the 
end  of  the  random  line.  If  the  end  of  the  random  line  docs  not 
agree  with  the  corner,  he  measures  the  distance  between  the  end 
of  the  random  line  and  the  true  corner.  This  distance,  multi- 
plied by  57.3,  and  the  product  'divided  by  the  length  of  the  line, 
will  give  the  correction  to  be  applied  to  the  assumed  declination. 

41.  Local   Attrac'i  n. — It   often   happens  that  ore   in   the 
ground,  a  wire  fence,  or  a  railroad  track,  etc.,  will  pull  the  needle 
out  of  the  magnetic  meridian.     When  this  is  discovered,  the  only 
thing  to   do   is   to   retrace   o;ir   steps   to  some  point   outside   the 
limits   of  the   attraction,   set  off  the  correct  bearing  and   locate 
some  point  ahead.     Then  transfer  to  it,  leaving  a  rear  flagman ; 
set  up  at  the  point  located,  and  sight  on  the  rear  flagman,  and 
then  prolong  the  line  by  locating  the  head  flagman,  transfer  an:l 
backsirht,    thus    locating   another  point.     This    method    will   not 
apply  when  the  whole  line  is  within  the  field  of  attraction.     We 
then  have  recourse  to  the  transit  and  locate  the  line  by  internal 
angles.     If  the  whole  farm  were  within  the  field  of  attraction,  it 
would  all  have  to  be  surveyed  with  the  transit  by  measuring  in- 
ternal angles. 

42.  Witnessing   a    Line    or    Corner. — All    corners    should 
have  witness  trees  or  some  natural  object  to  establish  the  corner, 
even  though  the  stake  disappears,  thus:     Begin  at  a  stake  from 
which  a  pecan  tree  10  ins.  in  diameter  marked  "K"  bears  S.  32° 
E.  84  varas.     To  find  the  corner  all  we  have  to  do  is  to  find  the 
witness   tree,   set  the  compass   at  it   on   the  reverse   bearing  and 
chain  off  the  distance.     As  a  check  it  is  well  to  have  the  witness 
line  at  the  corner  intersect  at  a  large  angle.     The  line  is  witnessed 
or  marked  by  line  trees.     All  trees  that  can  lie  reached  with  the; 
arm  either  way  by  a  man  standing  on  the  lint  should  be  marked 
with  three  hacks  on  the   side  next  to  the  line,  but  these  hacks 
should  not  cut  into  the  flesh  of  the  tree.     It  is  often  the  case  that 


36  SURVEYORS    HAND    BOOK. 

the  line  passes  through  a  tree ;  such  trees  are  marked  on  both 
sides  with  a  hack,  blaze,  hack.  These  trees  are  called  "fore  and 
aft"  trees.  If  trees  are  scattered  some  surveyors  hack  trees  that 
are  more  than  three  feet  from  the  line. 

43.  Typical  Field  Notes.— (From  Deed  Book  185,  p.  235, 
Travis  County,  Texas.)     Ber  inning  at  a  stake,  a  corner  to  H.  P. 
Sims   and    R.    D.    Rone,    from    which    a   hackberry    marked    "X" 
bears  N.  61°  30'  W.  10  varas ;  thence  N.  3°  W.  216  varas  to  a 
stone ;  thence   N.    16°   W.  255.6  varas   to  a  stone   from  which  a 
live  oak  marked  "A"  bears  N.  87°  W.  31  varas;  thence  S.  28°  30' 
W.  263.5  varas  to  a  stone  on  side  of  hill;  thence  S.  5°  16'  W. 
205.6  varas  to  a  stone,  from   which  a  pecan    12  ins.   in  diameter 
marked  T  bears  S.  63°  W.   18  varas  ;  thence  S.  856    15'  E.  227.3 
varas  to  the  beginning,  containing  —  acres  more  or  less. 

44.  Compass  Adjustments. — There  are  in  all   six    adjust- 
ments of  the  compass  that  should  be  made. 

First.  The  axis  of  revolution  should  be  perpendicular  to  the 
plane  of  the  plate.  This  is  done  by  the  maker  and  if  the  adjust- 
ment becomes  deranged,  the  instrument  should  be  sent  to  the 
maker  or  some  instrument  house  that  has  facilities  for  making 
such  repairs  or  adjustments. 

Second.  The  plane  of  the  plate  bubbles  should  be  parallel  to 
the  plane  of  the  plate.  If  the  first  adjustment  has  been  made, 
level  the  compass  and  then  turn  it  through  180°.  If  the  bubble 
remains  in  the  center  of  its  run,  no  adjustment  is  necessary. 
However,  if  the  bubble  does  not  stay  in  the  middle  of  its  run 
after  the  compass  has  been  turned  180°,  correct  half  the  apparent 
error  by  the  screws  at  the  end  of  the  bubble  tube.  Repeat  the 
operation  till  the  bubble  remains  in  the  middle  of  its  run  when 
the  compass  is  turned  180°. 

Third.  If  the  needle  is  bent,  its  ends  will  not  always  read  the 
same,  but  if  the  pivot  is  in  the  center,  the  difference  of  the  read- 
ings of  the  ends  will  be  constant.  To  straighten  the  needle,  set 
'one  end  at  zero  and  read  the  other  end.  This  reading  will  indi- 
cate the  way  the  needle  must  be  bent.  Repeated  trials  will  be 
necessary  before  the  needle  can  be  made  straight. 


COMPASS    SURVEYING.  37 

If  the  difference  of  the  readings  is  not  constant,  it  shows  that 
the  pivot  is  also  bent.  Read  the  ends  of  the  needle  in  any  posi- 
tion and  then  turn  the  needle  by  hand  till  the  north  end  is  in  the 
position  formerly  occupied  by  the  south  end.  Read  the-  south  end 
of  the  needle  and  note  the  difference  of  this  reading  and  the  first 
reading  of  the  north  end.  The  needle  can  then  be  bent  till  the 
north  end  when  swinging  free  will  bisect  the  space  between  the 
first  reading  of  the  north  end  and  the  second  reading  of  the  south 
end. 

Fourth.  If  the  pivot  is  bent  out  of  its  central  position,  the 
ends  of  the  needle  will  not  have  the  same 'readings,  and  the  dif- 
ference of  the  readings  will  be  variable.  After  the  needle  is 
straightened,  turn  the  compass  till  the  difference  of  the  end  read- 
ings is  the  greatest.  Remove  the  needle  and  bend  the  pivot  to- 
wards the  middle  of  the  larger  arc  that  was  between  the  ends.  Re- 
peat till  the  difference  of  the  end  readings  is  zero. 

Fifth.  The  plane  of  the  sights  can  be  made  normal  to  the 
place  of  bubble  tubes  by  leveling  the  compass  and  by  sighting  on 
some  plumb  line.  If  the  slot-sight  does  not  agree  with  the  plumb 
line,  the  base  of  sight  must  be  filed  till  a  plumb  line  can  be  seen 
throughout  the  sights. 

Sixth.  The  diameter  through  the  zero  graduations  should  be 
made  to  coincide  with  the  line  of  sights.  This  is  an  adjustment 
that  is  always  made  by  reputable  makers  and  the  surveyor  is 
rarely  called  upon  to  test  his  compass  for  this.  A  very  fine  wire 
stretched  through  the  sights  and  over  the  compass  box  will  indi- 
cate clearly  whether-the  line  of  sight  agrees  with  the  zero  lines. 

Bibliography. — "Davies'  Surveying.''  By  the  late  Charles 
Davies.  This  book  has  for  several  decades  been  one  of  the  stand- 
ards for  the  school  and  camp,  and  its  full  discussion  of  the  usual 
problems  of  land  surveying,  together  with  the  traverse  and  trig- 
onometric tables,  makes  it  a  valuable  assistant  to  the  surveyor  or 
a  guide  for  the  student.  In  addition  to  Davies',  the  works  of  the 
late  J.  B.  Johnson,  Wm.  G.  Raymond,  Breed  and  Hosmer,  etc., 
which  are  described  at  the  end  of  the  chapter  on  Transit  Sur- 
veying, contain  valuable  data  and  suggestions  for  the  compass 
surveyor. 


CHAPTER  III. 
TRAXS!  |  SURVEYING. 

45.  The  Transit.— The  essential  parts  of  a  transit  (Fig. 
24)  are,  mathematically,  a  line  of  sight  and  a  graduated  horizon- 
tal circle  for  reading  horizontal  angles.  Mechanically,  the  essential 
parts  are  the  telescope,  the  horizontal  axis,  the  circular  plates,  the 
spindle,  leveling  head,  tripod,  and  plumb-hob.  The  line  of  sight  is 
determined  and  defined  by  the  telescope  mounted  on  the  horizon- 
tal axis,  the  graduated  circle  by  a  horizontal  circular  plate  ui><  n 
which  the  degrees  and  fractions  of  degrees  are  marked  Tin- 
telescope  is  rigidly  attached  at  right  angles  to  two  horizontal  arms 
whose  axes  are  in  the  same  straight  line,  and  whose  outer  ends 
rest  in  the  standards.  These  standards  consist  of  two  diverging 
legs  rigidly  attached  to  the  horizontal  plate.  Two  small  levels  at 
right  angles  to  each  other  are  attached  to  the  horizontal  plate,  and 
by  means  of  these  the  plates  can  be  brought  to  an  absolute  hori- 
zontal. Two  verniers  (T  and  D,  Fig.  25,  are  attached  to  tin- 
plate  with  their  zeros  180°  apart  and  are  provided  with  a  gla-s 
<  \  er  for  protection.  These  verniers  are  turned  so  as  to  fit  the 
outer  graduated  circle  called  the  limb.  By  pulling  out  the  small 
clip  S  the  whole  upper  part  including  the  limb  R  can  be  taken  off 
the  head.  The  upper  part  of  the  transit,  including  telescope,  plate. 
horizontal  axis,  standards,  and  verniers,  is  called  the  alidade  and 
is  supported  on  a  spindle  and  can  be  turned  on  a  vertical  axis 
normal  to  the  vernier  plate.  However,  the  limb  B  and  alidade 
can  be  clamped  tight  together  by  a  clamp  DP  operated  by  the 
milled-hcad  screw,  which  is  seen  in  the  faint  outline  on  the  right 
of  Fig.  25.  When  clamped  the  alidade  and  limb  B  can  be  turned 
around  the  interior  spindle  H  by  unclamping  the  lower  clamp 
screw  (not  shown  in  Fig.  25  but  which  can  be  seen  in  Fig.  21 
The  transit  is  provided  with  a  level  head  as  in  the  Y-level,  which 
has  four  leveling  screws  for  bringing  the  limb  B  into  a  horizontal 
plane.  T'he  tripod  is  generally  made  of  light,  tough,  straight 
grained  wood,1  the  upper  ends  of  the  legs  being  connected  by 

38 


TRANSIT    SURVEYING. 


39 


40 


SURVEYOR'S    PIAND    BOOK. 


pin-joints  to  the  leveling  head,  while  the  lower  ends  are  shod 
with  metal  shoes.  The  plumb-bob  is  one  of  the  mechanical 
essentials  of  the  transit  as  the  instrument  cannot  be  set  over 
a  point  below  without  it. 

46.  Compass  Attachment. — Attached  to  and  supported  by 
the  upper  horizontal  plate  is  a  complete  compass  box,  includ- 
ing graduated  circle,  needle,  pivot,  a  declination  arc  inside  the 
box  and  under  the  needle.  The  declination  can  be  set  off 


P 

Fig.    25. 

and  the  bearings  read  as  in  the  compass,  and  the  telescope  simply 
helps  to  make  the  line  of  sight  more  exact.  However,  it  has 
the  disadvantage  of  having  its  line  of  sight  confined  to  a  single 
line,  which  a  leaf,  blade  of  grass,  etc.,  can  interrupt,  while  in 
the  compass  the  line  of  sight  is  confined  to  a  vertical  plane 
passing  through  the  slots  and  a  slight  interruption  to  the  line 
of  sight  can  be  obviated  by  moving  the  eye. 

47.     Vertical   Circle. — For  the  purpose  of  reading  angles 
of  elevation   a  vertical  circle  is  now  generally  attached  to  the 


TRANSIT    SURVEYING. 


41 


end  of  the  horizontal  axis  and  is  provided  with  a  tangent  screw 
and  a  vernier  reading  to  minutes.  It  is  not  an  essential  part 
of  the  transit.  To  bring  the  line  of  sight  to  a  horizontal  a 
bubble  tube  is  attached  to  the  telescope  whose  axis  is  made 
parallel  to  the  line  of  sight  of  the  telescope. 

48.  Shifting  Center.— The  modern  transits  are  furnished 
with  a  shifting  center.      The  lower  part  of  the  spindle  to  which 
the  loop  P  is  attached   works  in  a  ball  and   socket  joint  which 
is   extended    into   circular,   brim-like   plate   under    the   plate   on 
which  the  leveling  screws  rest.      If  these  are  loosened  so  that 
the  upper  part   of  the   transit  can  be  moved,   the  point  P  can 
be  moved  a  short  distance  in  any  direction.      This  is  called  the 
shifting  center. 

49.  The  Reticule. —  The  line 
of  sight  in  the  telescope  is  defined 
by  two  cross-wires  at  right  angles 
to  each  other,  cemented  into  de- 
pressions in  a  metal  ring,  Fig.  26. 
This   ring   is   inside   the  telescope 
and  is  controlled  and  operated  by 
four  capstan  screws  which  can  be 
seen  in  the  view  of  the  telescope 
of  the  level  or  transit.    The  whole 
arrangement  is  called  the  reticule 
and  it  is  susceptible  to  slight  mo- 
tions for  the  purpose  of  adjusting 

the  line  of  sight  of  the  telescope.  The  reticule  is  moved  by 
loosening  one  capstan  screw  and  by  tightening  the  opposite  one. 

50.  Setting  Up  the  Transit. — Set  up  the  tripod  with  the 
legs   widely  apart  and  firmly  pressed  into  the  ground;   take  the 
transit   out  of  the  box  by  taking  hold  of  the   limb  and  lifting 
the   entire  weight   with   one   hand,  simply  using  the  other  as   a 
guide      Never  grasp   the  transit  by  the  telescope  to  lift  it  out, 
as    such    lifting   springs    the    horizontal   axis    and   otherwise   in- 
jures   the    bearings.      Set    the    transit    on    the    tripod,    turn    it 
till  the  threads  catch,   revolve  the  telescope  vertically  and  take 
hold  of  two   legs   of  the  tripod  and  straighten  it  up   until  all 


4^  SURVEYOR'S    HAND    BOOK. 

the  legs  are  together,  and  then  place  the  tripod  across  the  shoul- 
der and  carry  it  to  the  place  where  the  observations  are  to 
be  made.  When  it  is  desired  to  set  the  tripod  over  a  point, 
place  the  legs  wide  apart,  and  move  them  so  that  the  plumb- 
bob  will  be  practically  over  the  point.  Level  up  the  instrument, 
and  if  the  plumb-bob  is  not  over  the  point  loosen  the  leveling 
screws  until  the  center  can  be  shifted,  then  move  the  center 
until  plumb-bob  comes  over  the  point  below  and  rclevel.  If 
there  is  not  sufficient  play  in  the  shifting  center  to  move  the 
plumb-bob  over  the  point  the  tripod  will  have  to  be  moved  in 
the  direction  necessary;  then  proceed  as  before. 

51.  Motions. — If  the  lower  clamp  screw  be  clamped,  and 
the  upper  loosened,  the  alidade  can  be  turned  on  the  vertical 
axis,  and  it  will  be  noticed   that  the  vernier  plate  moves  with 
the  alidade,  and  that  the  limb  or  graduated  circle  is  stationary. 
This  movement  is  called  the  upper  motion.     If  the  upper  clamp 
screw   be   tightened   and  the   lower  one   loosened,   that   part  of 
the   instrument  above  the  leveling  head  can  be   turned  around 
one  of  the  spindles.     This  movement  is  called  the  lower  motion. 

52.  Use  of  the  Transit.— After  the  transit  has  been  set 
up  over  a  point,  make  the  zero  of  the  vernier  agree  with  the 
zero   of   the    limb.     Unclamp    the    upper    motion   and   bring    the 
two   zeros   as  near  together  as  possible;   then   clamp  the   upper 
motion    and    bring   the    zeros    into    exact    coincidence   by   means 
of   the   tangent   screw   controlling  the  upper    clamp.     After   the 
zeros    have    been    brought    together,    loosen    the    lower     motion 
clamp,  take  hold  of  the  limb  with  both  hands  and  turn  the  tele- 
scope   till    it   points    towards    the    object    on   which   we   wish   to 
observe.     The   telescope  can  be   brought  approximately  into   the 
required   direction  by  sighting   over  the  telescope  at  the  object 
and   turning   the   instrument    until   the   telescope  points   towards 
the  point.     The  cross-wires  are  brought  into  the  field  of  view  by 
turning  the  screw  that  operates  the  eye-piece.     The  large  milled- 
head  screw  on  side  of  the  telescope  is  then  turned  till  the  ob- 
served object  is  seen  distinctly  and  clearly  through  the  telescope. 
The  tangent  screws  can  then  be  turned  till  the  vertical  wire  bi- 
sects the  object. 


TRANSIT    SURVEYING.  43 

53,  The  Transit  as  a  Compass. — If  it   is   desired  to  use 
the  transit  as  a  compass   in   regular   surveying   work,   or   to  use 
the   needle   as    a   check   on    other   work,   the   milled-head   screw 
shown   on  the   outside  of  the  left  leg  of  the   front  standard  in 
Fig.    24    is    loosened,   and   the    milled-head    screw    that    controls 
the  declination   arc,  seen  betwreen  the  rear  standards,   is   turned 
until  the  proper  declination  is  set  off  by  the  vernier  inside  the 
compass  box.     These  screws  are   then  clamped   and   the   transit 
will    then    read    angles   with    the   true   meridian.     The   needle    is 
turned   loose  by  means   of   the  milled-head   screw   shown   above 
the  plate  on  the   right  of  Fig.  24. 

54.  Transit   Surveying.— If  the   transit   is  not   used  as  a 
compass,  we  must  read  the  azimuth  of  each  course  or  line  in- 
stead of  the  bearing.     As  this  azimuth  is  read  from  the  south 


point  around  by  the  west,  north  and  east,  and  on  to  the  south 
again,  we  can  have  with  a  transit  reading  to  minutes,  an  azi- 
muth of  3o9°  59',  -which  could  be  a  bearing  of  S.  0°  01'  E. 
These  azimuths  are  read  with  reference  to  the  true  meridian 
and  it  is  necessary  to  locate  this  very  accurately  if  the  abso- 
lute azimuth  is  desired.  However,  if  it  is  only  an  accurate  ex- 
pression for  the  area  of  the  farm,  a  meridian  can  be  assumed 
for  the  first  course,  and  then  carried  around  the  farm  by  lo- 
cating this  meridian  from  each  course. 

55.  Transit  Vernier.— The  transit  vernier  is  a  double  ver- 
nier (Fig.  27)  and  has  30  divisions  on  each  side  of  its  zero. 
Each  half  of  the  vernier  covers  29  parts  or  divisions  on  the 
limb.  The  smallest  division  on  the  limb  is  a  half-degree,  or 
thirty  minutes,  and  hence  the  vernier  can  read  to  one-thirtieth 
of  a  half  degree  or  to  one  minute.  The  angle  may  be  meas- 


44  SURVEYOR'S    HAND    BOOK. 

tired  from  the  right  or  left,  and  the  one  we  use  depends  upon 
the  special  problem  under  consideration.  If  read  from  the 
right  we  see  that  the  zero  of  the  Vernier  is  between  5°  30' 
and  6°  00'.  The  reading  is  5°  30'  plus  the  vernier  reading.  As 
the  reading  of  the  transit  is  from  the  right,  use  the  left  half 
of  the  vernier.  On  examination,  w^e  find  that  the  14th  di- 
vision of  the  vernier  agrees  with  a  division  mark  on  the  limb. 
The  vernier  reading  is  therefore  14'.  The  whole  angle  reading 
is  therefore  5°  44'. 

If  the  angle  is  to  be  read  from  the  left,  use  the  vernier  on 
the  right.  The  zero  of  the  vernier  lies  between  354°  and  ?»")4° 
30'.  The  16th  division  of  the  vernier  on  the  right  agrees  with 
a  division  on  the  limb  and  the  vernier  reading  is  therefore 
16',  and  the  whole  angle  reading  is  354°  16'. 

56.  Example. — If  a  farm  is  surveyed  with  the  transit,  the 
field  notes  would  be  as  follows: 

Course.  Azimuth.  Distance. 

AB 203°  30'  255.72  varas 

BC 248°  00'  .                         182.10  varas 

CD 3°  47'  329.42  varas 

DA.... 84°  15'  249.92  varas 

It  will  be  observed  that  the  shape  and  dimensions  of  the 
farm  would  not  have  been  changed  in  the  slightest  if  the  first 
course  AB  had  been  taken  at  202°  instead  of  203°  30'.  It  simply 
would  have  amounted  to  a  turning  of  all  meridians  in  a  clock- 
wise direction  and  the  azimuths  would  have  been  as  follows : 
202°,  246°  30',  5°  17',  82°  45'.  Then,  if  it  is  desired  to  obtain 
the  area  accurately,  we  can  assume  a  meridian,  and  it  is  not 
necessary  that  this  be  the  true  meridian,  but  when  this  me- 
ridian is  once  assumed,  the  azimuth  of  all  the  courses  must 
be  with  reference  to  it. 

57.  Reference  Lines. — The   line  to  which   the  azimuth  is 
referred  can   be  assumed   in   any   desired   direction,  and  one   of 
the  sides  is  often  taken  as  this  reference  line  if  only  the  area 
is   required.     Thus,   in   the    example,    if  AB    is    assumed   as    the 
reference  line,  the  azimuths  with   respect  to  this   line  are   180°, 
224°  30',  340°   17',   60°  45'.     In  calculating  the  area  the  bear- 


TRANSIT    SURVEYING.  45 

ing   can    be    taken    with    respect    to    the    reference   line.     If   AB 
were  our  reference  line,  the  field  notes  would  be  as  follows  : 
Course.  "    Bearing.  Distance. 

AB North  255.72  varas 

BC N  44°  30'      E  182.10  varas 

CD S  19°  43'      E  .     329.42  varas 

DA N  60°  45'    W  249.92  varas 

PROBLEM  21.— In  a  farm  ABCDE,  ,45  =  19.90  chains,  BC  — 
9.03  chains,  CD  =  9.77  chains,  #£=5.67  chains,  £,4  =  13.24 
chains;  A=8»c  12',  5  =  73°  37',  C  =  139°  08',  D  =  l(>3°  40',  £ 
=  74°  24'. 

If  the  azimuth  of  AB  =  18Q°,  find  the  azimuth  and  the  bear- 
ings of  the  other  lines. 

58.  Repeating  Method. — It  is  often  desired  to  find  the 
angle  more  accurately  than  it  can  be  read  by  a  single  reading 
of  the  verniers.  If  we  have  a  transit  reading  by  vernier  to 
one  minute,  we  can,  find  any  angle  ABC  to  any  desired  fineness 
by  the  repeating  method.  Thus,  if  the  one  transit  verniers 
read  to  one  minute,  we  can  find  the  angle  to  ten  seconds  by 
repeating  the  observation  six  times.  The  process  is  as  follows : 
Telescope  normal : 

1.  Set  transit   on  point  B,  level   up  and   set  cross-wires   on 
point  A   and   read  both  verniers. 

2.  Unclamp  upper  motion  and  deflect  to  C,  clamp  upper  mo- 
tion  and  read  both  verniers. 

3.  Unclamp  lower  motion,  deflect  to  A  and  clamp. 

4.  Unclamp  upper  motion  and  deflect  to  C  and  clamp. 

5.  Unclamp  lower  motion,  deflect  to  A,  and  set  thereon. 

6.  Unclamp  upper  motion,  deflect  to  C ' ,  set  cross-wires  there- 
on  and   read  the  angle  as  given  by  both  verniers.     This  result 
is  three  times  the  angle  ABC,  etc.,  etc.,  etc. 

The  process  can  be  carried  on  till  there  have  been  five, 
ten,  or  twenty  deflections  by  upper  motion  from  A  to  C,  thus 
measuring  the  an^le  five,  ten,  or  twenty  times.  Both  verniers 
should  be  read  in  every  case  and .  the  average  taken.  Usually 
the  angle  is  read  a  given  number  of  times,  as  above,  with  the 
telescope  normal  beginning  right  (or  left)  station  A,  and  then 


46  SURVEYOR'S    JIAXD    BOOK. 

read   the    same    number    of    limes    with    the    telescope    reversed, 
beginning  on  the  left    (or  right)    station  C. 

Example.  Vernier  A.  Vernier  /< 

1 : 31°  42'  31°  .!•_>' 

3 95°  07'  95°  07' 

5 158°  31'  158°  32' 

The  average  of  the  five  readings  gives  158°  31'  30",  or  an 
angle  of  31°  42'  18". 

59.  To  Adjust  the  Plate  Levels. — The  axis  of  the  plate 
levels  should  be  at  right  angles  to  the  vertical  axis  or  the  axis 
of    revolution.     Set    the    transit    up    on    the    tripod,    level    it    by 
the  plate  levels  as  near  as  possible,  bring  one  of  the.  level  tubes 
parallel   to   a  pair  of  leveling   screws,   and  bring  the  center  of 
the   bubble    exactly    to   the    center    of   its    run.     Then   tarn   the 
alidade  180°  on  its  vertical  axis,  and  if  the  bubble  remains  in 
the  center  of  its  tube,  it  is   in   adjustment.     If   not,   lower  the 
high  end  of  the  tube  or  raise  the  low  end  by  means  of  the  small 
capstan   screws   at   the  end  of  the  tube   a   sufficient  amount  to 
correct  half  of  the  displacement  of  the  bubble.     Correct  the  re- 
mainder by  means  of  the  leveling  screws  and  repeat  as  a  check 
on  your  work.     Usually  it  takes  several  trials  to  make  this  ad- 
justment. 

60.  Line    of    Sight    Adjustment. — To    make    the    line    of 
sight    perpendicular    to    the   horizontal   axis,    set    up    the    instru- 
ment on  some  plane  nearly  level,  bring  the  plate  bubbles  to  the 
center  of  their  run,  and  locate  a  point  about  100  to  200  ft.  from 
the  instrument; 'turn  the  instrument  on   its  horizontal  axis  and 
locate  another  point  the  same  distance  from  the  instrument,  but 
in  an  opposite  direction ;  revolve  the  alidade  and  bring  the  ver- 
tical wire  in  coincidence  with  the  point  first  located ;  then  turn 
the    telescope    on    its    horizontal    axis    and    locate   another   point 
near  the  second  point  located  in  the  intersection  of  the  cross- 
wires.     If  this  point  last  located  coincides  with  the  second  point 
located,  the  line  of  sights  is  perpendicular  to  the  horizontal  axis. 
If  it  is   not,   correct  one-fourth   of  the  displacement  and   mark 
this  point,  and  proceed  as*  before. 

Let  AB,  Fig.  28,  be  the  position  of  the  horizontal  axis  when 
the  point  1  is  located,  and  let  the  line  of  sights  make  an  angle 


TRANSIT    SURF  EYING.  47 

of  10.r  =  dr  with  the  perpendicular  xy  to-  the  axis.  Revolve  the 
telescope  on  the  horizontal  axis  and  locate  the  point  2.  Now  the 
angle  102=180° — 2u.  Turn  the  alidade  around  the  vertical 
axis  till  the  line  of  sights  intersect  1.  As  02  has  been  turned 
through  the  angle  102  or  180°—  2a,  AB  has  been  turned  through 
the  same  angle  and  occupies  the  position  A'B',  where  AO'A  = 
180° — 2o,  orA'OB  =  2a.  The  perpendicular  has  moved  to  the 
position  of  x'y'  where  xOx'  =  180°  —  2a  or  x'Oy  =  2a.  Let 


telescope  point  to  1,  then  revolve  it  on  its  horizontal  axis 
and  locate  a  point  3  in  the  line  of  sights  near  2.  The  angle 
103  =  180°— 2o.  Therefore,  the  angle  203  =  4o;  hence,  all  we 
have  to  do  is  to  bring  the  line  of  sights  into  coincidence  with 
x'y'.  Divide  the  angle  203  into  four  parts  and  make  30,r'  — j, 
one-fourth  of  the  angle  302.  We  can  do  this  by  setting  a  point 
x'  at  one-fourth  of  the  distance  from  3  to  2,  and  as  02  and  03 
are  several  hundred  times  2-3,  this  is  as  accurately  as  we 
can  measure  the  angle  30;r'  equal  to  one-fourth  of  203.  Now, 
keep  the  axis  clamped  in  the  position  A'B' ,  and  move  the  ver- 
tical wire  by  capstan  screws  till  it  coincides  with  the  point  x '. 
Repeat  whole  work  until  it  checks. 


48 


SURVEYOR'S    HAND    BOOK. 


61.  Peg  Adjustment. — The  axis  of  the  bubble  tube  may 
be  made  parallel  to  the  line  of  sights  by   the  peg  adjustment. 
Drive  two  pegs   or   stakes    in   the   ground   about   200   ft.   apart, 
whose  difference  of  level  is  less  than  4  ft.     Set  the  transit  mur 
peg  A,  level  the  instrument,  and  turn  the  telescope  so  that  the 
eye  end  is  over  the  peg,  while  the  bubble  is  in  the  center  of  its 
run;  measure  the  height  of  the  center  of  the  eye-piece  above  the 
peg  and  call   this  distance  h.     Have  an  assistant  hold   the   rod 
on  top  of  peg  B  and  measure  from  where  the  line  of  sights  cuts 
the  rod  to  the  top  of  peg  B  and  call  this  r.    Transfer  the. transit 
to  peg  B   and    set   up   as  before,  measuring  the   height  of  the 

center  of  the  eye-piece  from 
the  top  of  peg  B  and  call 
this  distance  h'.  Have  the 
rod  placed  on  top  of  peg 
A  and  measure  the  distance 
from  the  line  of  sights  to 
the  top  of  peg  A.  and  call 
FI*-  29-  this  distance  r. 

In  Fig.  29,  AK,  CD  and  FG  are  the  horizontal  lines  as  de- 
termined by  the  bubble  tube.  Suppose  the  line  of  sights  C/T 
cuts  below  the  horizontal  line  an  amount  of  DE  =  e',  when  the 
transit  is  transferred  to  B  it  will  again  cut  below  when  the 
telescope  is  sighted  to  A  an  amount  HG  =  e. 

Let  AC  =  h    BE  =  r,  BF  =  h'^lH=rf.     Then  the  true-dif- 
ference of  level  of  A  and  B  =  BK  =  BD  —  AC  =  r  +  c  —  h. 
Also    BK  =  BF  —  AG  =  h'—(r'  +  e) 
Therefore  r  +  e  —  h  =  h'  —  r'  —  e 
Therefore  <•=%  [(h  +  //')  —  (r  +  r')]. 

Rule:  The  double  error  is  equal  to  the  sum  uf  the  instru- 
ment heights  minus  the  sum  of  the  rod  heights. 

62.  Location  of  Meridian  by  Polaris. — Table  I  gives  the 
times  when  Polaris  and  the  mean  sun  are  on  the  meridian  to- 
gether.    For  1907  the  "epoch"  is  14.1     This  means  that  the  mean 
sun   and   Polaris   are  on  the  meridian   together  April    14,   one- 
tenth   of   a    day   after   the   beginning   of   April   14 — that   is,   2.4 
hours  after  the  beginning  of  April   14.     This  would  make  the 


TRANSIT    SURVEYING.  49 

"epoch"  occur  on  April  14  at  2:24  A.  M.  For  1909  the  "epoch" 
is  13.8,  or  April  13,  7:12  P.  M.  The  "epoch,"  then,  is  the  time 
or  date  when  Polaris  and  the  mean  sun  are  on  a  meridian  at 
the  same  time. 

Table   I — Epochs    equal   date   in   April   when    Mean    Sun   and 
Polaris  are  on  a  Meridian  together: 
Year.  Epoch.  Year.  Epoch.  Year.  Epoch.  Year.  Epoch.  Year.  Epoch. 

1907  14.1   1912    13.9    1917    14.6    1922    15.3    1927    15.9 

1908  13.5    1913    14.2    1918   15.0    1923   15.6   1928    15.3 

1909  13.8   1914   14.6   1919   15.3    1924    15.0    1929   15.6 

1910  14.2    1915   14.9    1920    14.7   192.'    15.3   1930    15.9 

1911  14.5   1916   14.3    1921    15.0   1926   15.6    1931    16.2 

If  Polaris  and  the  mean  sun  are  on  a  meridian  together,  the 
mean  sun  will  reach  the  meridian  4  minutes  later  than  Polaris  on 
next  day. 

The  hour  angle  of  the  star  will  be  more  than  that  of  the 
sun  by  3.5)4  multiplied  by  the  number  of  days  after  the  epoch. 

EXAMPLE:  Find  the  position  of  the  star  (/)  in  its  orbit 
at  9  P.  M.  May  6,  1907.  The  "epoch"  for  1907  is  on  April  14 
at  2:24  A.  M.  The  number  of  days  from  2:24  A.  M.,  April  14, 
to  9  P.  M.,  May  6,  is  22.775.  Hence  Polaris  will  be  22.775  X 
3.94  =  89.73  min.  ahead  of  the  sun.  At  9  P.  M.  the  sun  is  9 
hrs.  past  the  meridian  of  the  observer,  hence  Polaris  will  be 
0  hrs.  plus  89.73  min.  or  10  hrs.  29.73  min.  past  the  meridian. 
By  using  this  time  (/)  in  Table  II  we  can  find  the  angle  Po- 
laris makes  at  that  time  with  the  true  meridian. 

Table  II.— /  =  local  mean  time  +  3.94  (date— epoch). 
Hours. 

t  Angle,  a.  Lat.  cor.,  b.  t 

0  0'  —74'  24 

1  25'  —72'  23 

2  49'  —64'  22 

3  69'  —52'  21 

4  84'  —37'  20 

5  93'  —19'  19 

6  96'  0  18 

7  92'  +19'  17 

8  82'  +36'  16 

9  67'  +51'  15 

10  47'  +63'  14 

11  24'  +70'  13 

12  0'  +72'  12 


1900. 

1910. 

1920. 

1930. 

.82 

.78 

.75 

.72 

.88 

.85 

.81 

.77 

1.00 

•   .96 

.92 

.87 

1.10 

1.14 

1.09 

1.04 

50  SURVEYOR'S    HAND    BOOK. 

Table  III. — Azimuth   Coefficients. 

Coefficients  =  K. 
Lat. 
20° 
30° 
40° 
50° 
Table  IV. — Lat.  correction  Coefficient. 

Year.  Coefficient,  Q. 

1900  1.00 

1910  .!)<» 

1920  .92 

1930  .87 

EXAMPLE:  Find  the  angle  that  Polaris  makes  \\ith  the  true 
meridian  at  9  P.  M.  May  6,  1907,  in  latitude  3o.  The  time  in- 
terval from  epoch  to  date  was.  22.775  days  and  the  increase  in 
time  was  1  hr.  and  29.77  mins.  The  value  of  t  was  found  to 
he  10  hrs.  and  29.73  mins.  or  10.50  hrs.  (which  is  near  enough 
for  our  purposes).  Looking  in  Table  II  under  /  for  1  •).-">  ho'r.s 
we  find  that  we  have  to  interpolate  between  47'  and  21',  hence 
the  angle  is  35.5'.  This  must  be  multiplied  by  the  a/imuth 
coefficients.  For  1900,  lat.  30,  the  coefficient,  Table  III,  is  0.88, 
and  for  1900  it  is  0.85.  For  one  year  the  decrease  is  0.003,  and 
for  seven  years  it  is  0.021.  The  coefficient  is  therefore  0.80.  The 
angle  or  azimuth  with  the  north  meridian  =35. 5  X  .86  =30.6' 
west.  The  observed  altitude  of  the  star  was  29°  8'.  The  lati- 
tude coefficient  for  1907  lies  between  1.00  and  0.96  and  an  in- 
terpolation gives-  .972.  From  Table  II,  lat.  cor.  (b)  is  00.5'. 
Hence  the  correction  for  the  altitude  will  =  .972  X  60.5  =  04.04'. 
The  latitude  =29°  8'  +  64.04'  =  30°  12.64'. 

PROBLEM  22. — Find  the  angle  that  Polaris  makes  with  the 
true  meridian  9  P.  M.  June  12,  1907,  in  latitude  of  33°.  An- 
swer =20'  east. 

PROBLEM  23. — Given  latitude  of  place  =  36°,  find  the  angle 
Polaris  makes  with  meridian  on  November  6,  1909,  10  P.  M. 
Answer  =  9'.2  east. 

PROBLEM  24. — An  observation  was  made  on  Polaris  at  9:30 
P.  M.  July  22,  1908,  in  latitude  30°.  Find  the  angle  made  with 
the  meridian.  Answer  =71'  east. 


TRANSIT    SURVEYING. 


51 


NOIlVNIINYu X3MQ1 


* 


*  >  Delta 


63.  Circumpolar  Stars. — A  meridian  can  be  located  with 
sufficient  accuracy  for  ordinary  surveying  by  observations  on  the 
North  Star  (known  as  Polaris),  which  is  about  one  and  one- 
fifth  degrees  from  the  true  North  Pole,  and  if  we  would  ob- 
serve it  for  a  whole  day  it  would  appear  to  describe  a  circle 
about  the  North  Pole  in  a  direction  contrary  to  the  motion  of 
the  hands  of  a  clock,  /.  c.,  contra-clockwise.  On  account  of  the 
invisibility  of  the  true 
North  Pole  this  motion  can 
be  best  observed  by  select- 
ing some  star  in  the  Dipper. 
If  we  could  noce  exactly 
when  one  of  the  stars  of  the 
Dipper  is  directly  above 
Polaris  and  could  follow  its 
motion  throughout  the  bal-  ^ 
ance  of  the  night,  the  next  ^ 
day  and  part  of  the  next  I- 
night,  we  would  observe  |- 
that  the  star  would  again  ^ 
reach  a  point  directly  above  || 
Polaris  four  minutes  earlier 
than  it  did  on  the  preced- 
ing day.  If  we  observed  it 
directly  above  Polaris  at  10 
P.  M.  on  one  night,  the  next 
night  it  would  be  at  the 
same  position  at  56  mins. 
after  9  o'clock.  Thus,  each 
of  these  stars  gains  four 
minutes  each  day  (exactly,  3.945  minutes).  In  one  year  it 
would  gain  24  hours  and  would,  therefore,  make  one  more  revo- 
lution than  the  earth  makes  on  its  axis.  All  stars  that  make 
an  apparent  complete  revolution  about  the  North  Pole  are  called 
circumpolar  stars,  and  any  of  them  could  be  used  for  the  location 
of  a  meridian  when  the  selected  star  is  directly  above  or  below 
Polaris. 


Polaris 
Pole 


Major  /     \ 

' 


'The  Dipper 


UPPER  CULMINATION 
Fig.    30. 


SURVEYOR'S    HAND    BOOK. 


There  are  two  groups  of  stars  (called  constellations)  sit- 
uated opposite  to  each  other  with  resp.ect  to  Polaris  and  the 
North  Pole  that  afford  favorable  opportunity  for  the  location  of 
the  meridian  by  surveyors.  These  constellations  are  those  of 
the  Great  Bear  (the  Dipper)  and  of  Cassiopeia  (the  Chair).  By 
a  glance  at  the  outlines  of  these  constellations,  Fig.  30,  it  will 
be  seen  that  the  dotted  lines  outline  the  shape  of  a  dipper  and 
chair,  respectively,  hence  the  names.  It  must  be  remembered 
that  Polaris  is  always  opposite  the  Dipper  with  respect  to  the 
pole,  and  that  it  is  on  the  same  side  as  the  Chair. 

When  a  star  is  directly  above  the  pole  it 
is  said  to  be  at  its  up  per  culmination,  and 
when  directly  below,  at  its  lower  culmina- 
tion. When  at  the  eastern!  point  of  its  orbit, 
it  is  said  to  be  at  its  eastern  elongation,  and 
when  at  its  western  point,  at  its  western 
elongation. 

64.  Location  of  Meridian. — A  line 
passing  through  the  second  star  (Zeta)  in 
the  handle  of  the  Dipper  and  the  third  in 
the  back  of  the  Chair  (Delta  Cassiopeia) 
passes  through  Polaris  and  the  North  Pole. 
When  Zeta  of  the  Dipper  or  Delta  Cas- 
siopeia is  directly  above  or  below  Polaris, 
Polaris  is  on  the  meridian,  and  is  at  its 
upper  or  lower  culmination.  If  the  Dip- 
per is  above,  Polaris  is  below  the  pole,  and  vice  versa.  But  when 
the  star  is  at  either  culmination,  its  horizontal  motion  is  more 
rapid  than  at  any  other  point  in  its  path,  and  a  slight  error  in 
time  affects  the  result.  When  the  star  is  at  either  elonga- 
tion, the  direction  of  its  motion  is  vertical,  and  a  slight  error 
in  time  does  not  have  such  decided  influence  on  the  azii.iuth. 
65.  PZS  Triangle.— The  North  Pole  (P),  the  Zenith  (Z) 
and  the  Sun  (S)  form  a  spherical  triangle  PZS,  Fig.  31,  where  if 


Fig.    31. 


TRAXS1T    SURVEYING.  53 

/=  latitude  of  observer 
£  —  hour  angle 
a  —  azimuth  of  sun 
d  =  declination  of  sun 
h  —  altitude  of  sun. 
We  have  : 

PZ  =  co-latitude  =  00  —  /  ; 
PS  =  co-declination  —  90  —  d  ; 
ZS  =  co-altitude  =  f  0  —  h  ; 
ZPS  —  hour  angle  of  sun  —  £; 
SZM  =  azimuth  360  —  a. 

66.  Formulas.  —  The  usual  problem  is  to  locate  a  meridian 
at  a  certain  place  whose  latitude  and  longitude  are  known.  Drop 
a  perpendicular  from  6*  on  the  meridian  PZ  of  the  observer, 
cutting  it  at  M,  and  let  ME  =  N  where  EQ  is  the  celestial  equa- 
tor, or  earth's  equator  extended  to  the  heavens. 
Now,  ZE  —  latitude  —  /. 

:.ZM  —  \  —  N. 

By  the   application   cf   Napier's   Tangent   Law,  we  have,   in  the 
right  triangle  PSM: 

cos  f=tan  PM  cot  PS 
cos  t  =  tan    (90—  AO    cot   (90  -d) 
—  cor  N  tan  d 
tan  d 


In  the  right  triangle,  SZM 

sin  ZM=-tan  MS  cot  a  or  tan  l\fS:=siii  ZM  tan  a. 
In  the  right  triangle  MPS, 

sin   PM  =  tan  MS  cot  t  or  tan   MS  =  sin  PM  tan  t 
Equating  the  two  values  of  tan  MS  ,  we  get  : 

tan  a  sin  ZM  =  sin  PM  tan  t 

sin  PM  tan  t 
tan  a  *         sin  ZM 
But 

PM  =  M  —  N,   ZM  =  l  —  N 
cos  N  tan  t 


sin(l-N) 


54  SCKl'liyOK'S    //.LVD    BOOK. 

67.  Observation  on  Sun. — The  best  time  of  day  to  make 
an  observation  for  azimuth  on  the  sun  is  from  c  to  10  A.  M. 
and  from  3  to  o  P.  M.  Before  an  observation  -.s  made  it  is 
necessary  to  have  mean  local  time  and  if  a  chronometer  is  not 
available,  two  watches  should  be  set  to  agree  with  Western 
Union  time.  Thirty  minutes  before  the  observation  is  to  be 
made  the  transit  should  be  set  over  the  station,  the  verniers 
should  be  brought  to  zero,  and  the  transit  be  pointed  to  some 
definite  terrestrial  point,  as  a  church  spire.  The  transit  should 
then  be  turned  by  upper  motion  to  point  approximately  at  the 
sun,  and  as  soon  as.  the  sun  comes  into  the  field  of  view  of  tin- 
telescope,  the  observer  should  clamp  the  upper  motion  and  call 
"angle,"  when  two  men  read  the  angles  as  given  by  the  two 
opposite  verniers.  At  the  signal  "angle"  the  timekeepers,  of 
which  there  should  be  at  least  two,  get  ready  to  observe  the 
time.  As  the  disc  of  the  sun  approaches  the  vertical  wire,  the 
observer  calls,  "Get  ready,"  and  just  as  the  edge  of  the  sun's 
disc  coincides  with  the  vertical  wire  he  calls  "time"  and  im- 
mediately moves  the  vertical  wire  by  aid  of  the  tangent  screw 
till  the  opposite  edge  of  the  sun's  disc  coincides  with  vertical 
wire,  when  he  calls  "time*'  again.  The  time  interval  between 
the  two  calls  of  "time"  should  not  be  over  six  seconds.  The 
timekeepers  have  noted  both  the  hours,  minutes  and  seconds 
at  each  call  of  "time,"  and  the  angle  readers  read  both  angles 
and  record  same.  The  data  taken  in  the  field  therefore  con- 
sist of  reading  the  spire-station-sun  angle  for  both  discs  of 
sun,  and  the  times  corresponding  to  these.  The  average  of  each 
is  taken  as  the  angle  and  time  of  the  sun's  center  The  local 
mean  time  is  reduced  to  apparent  time,  and  this  to  degrees, 
which  gives  the  hour  angle. 

The  declination  of  the  sun  is  found  for  the  given  time 
and  N  is  found  from  Formula  5,  and  the  substitution  of  values 
of  N,  t  and  /  in  Formula  6  will  give  the  angle  a. 

The  second  method  of  finding  the  angle  a  consists  in  meas- 
uring the  altitude  of  the  sun  at  the  time  of  observation.  To 
do  this,  the  disc  of  the  sun  is  brought  to  tangency  with  the 
vertical  wire  and  on  its  left,  so  that  the  lower  edge  of  disc 


TRAXSIT    SURVEYING.  -V> 

coincides  with  the  horizontal  wire.  If  we  regard  the  cross-wires 
as  axes,  the  sun  would  be  in  the  second  quadrant  and  tangent 
to  both  axes  at  the  first  observation.  In  this  position  we  record 
time,  the  spire-station-sun  angle,  and  the  vertical  angle.  The 
disc  is  then  brought  into  the  fourth  quadrant,  so  that  it  touches 
the  two  axes,  when  the  same  data  are  observed  as  before.  The 
average  of  these  is  taken  as  the  spire-station-sun  angle,  the 
angle  of  elevation,  and  the  time  of  observation.  Then  the  angle 
is  corrected  for  refraction  and  this  gives  us  the  complement  of 
ZS  of  the  triangle  PZS.  The  three  sides  of  the  triangle  PZS 
are  thus  known  whence  the  angle  PZS  can  be  calculated. 
Let*  =  i(PZ  +  ZS  +  PS). 

I  ~sin  (s—PZ)  sin  (s—ZS) 
Then  sin  \PZS  =  -J          sin  pz  sin  ZS 

68.  Refraction. — The    effect   of   refraction    is    to    raise   all 
bodies   and   make  them  appear  higher  than  their  true  positions. 
Thus  the   sun   can   be   seen  wholly  above   the  horizon,   when  in 
reality   no  part  of   it  is  above.     If  R   represents   the  amount  of 
refraction  in  seconds   of  arc  and  h  is  the  altitude    of  the    sun, 
we  have: 

R  =  r>8"    tan    h 
Table   V. — Table  of  Refractions  : 

Elevation.    Refraction.   Elevation.    Refraction.    Elevation.    Refraction. 

5J  9'  52"  16  3'  20"  35  1'  23" 

10°  5'  19"  1V  3'  08"  40  1'  09" 

11°  4'  51"      '  18  2'  58"  45  0'  58" 

12°  4'  28"  19  2'  48"  50  0'  49" 

13°  4'  07"  20  2'  39"  60  0'  34" 

14°  3'  50"  25  2'  04"  70  0'  21" 

15°  3'  34"  30  1"  41"  80  0'  10" 

69.  Solar  Attachment. — There  are  various  forms  of  solar 
attachments,    but    we    shall    here    describe    only    two.       Fig.    32 
shows  a  diagonal  prism,  which  "consists  of  a  prism  attached  to 
the   cap   of   the   eye-piece,   by  which   the  object  is  presented   to 
the    eye    when   placed    at   right   angles    to    the   telescope.     When 
the  telescope  is  directed  to  the  sun  the  slide  or  darkener  con- 
taining the  colored  glass  is  moved  over  the  opening.     The  cir- 
oilar  plate  with  which   the  prism  is  connected  is  made  to  turn 
in  the  cap,  so  that  when  it  is  substituted  for  the  ordinary  cap 
of  the  eye-piece  the  opening  of  the  prism  can  be  easily  adjusted 


56  SURVEYOR'S    HAND    BOOK. 

to  the  position  of  the  eye.     Observations  can  be  taken  with  the 
prism  up  to  an  angle  of  60°  of  elevation." 

The  other  form  of  solar  attachment  consists  of  a  second 
telescope,  generally  smaller  in  size,  attached  to  the  regular  tele- 
scope of  the  transit.  The  second  telescope  is  provided  with 
colored  glass  to  enable  the  observer  to  see  the  sun  with  dis- 
tinctness and  definition.  Fig.  33  illustrates  a  common  form  of 
this  solar  attachment  which  is  provided  with  telescope  level 
.-UK!  tangent  screws  for  horizontal  and  vertical  motions.  The 
line  of  sight  of  the  solar  telescope  cin  be  made  parallel  to  that 
of  the  transit  by  bringing  both  bubble  tubes  to  the  middle  of 
their  run,  while  the  telescopes  are  pointed  at  a  vertical  line 
some  '200  ft.  away.  This  line  should  be  marked  on  a  white 
sheet  of  paper  tacked  to  the  side  of  a  house 
on  the  same  level  practically  with  the 
telescopes.  Draw  two  heavy  horizontal 
lines  on  this  sheet  of  paper  at  a  distance 
apart  equal  to  the  distance  between  the 
axes  of  the  telescopes.  .Bring  the  cross 
wires  of  the  transit  telescope  on  the  lower, 
of  these  lines,  and  if  the  lines  of  sights  arc 
parallel  the  line  of  sight  of  the  solar 
telescope  will  intersect  the  upper  horizontal  line.  If  it  docs 
not,  adjust  its  reticule  till  the  line  of  sight  as  defined  by  the 
cross-wires  intersect  the  upper  line.  Check  till  perfect  agree- 
ment is  secured.  An  error  of  1-16  in.  in  the  distance  between 
the  axes  in  200  ft.  would  produce  an  error  in  the  parallel  align- 
ment of  the  lines  of  sight  of  dnly  5".  A  longer  base  would 
reduce  the  error.  If  the  base  is  507  ft.  and  the  error  in  dis- 
tance between  axes  is  1-30  in.,  the  lines  of  sight  will  make  an 
angle  of  1". 

To  eliminate  light  errors  in  latitude  and  as  a  check  on  the 
work,  observations  can  be  taken  in  the  forenoon  and  after- 
noon at  about  the  same  time  from  the  meridian  passage  of  the 
sun.  In  each  set  of  observations  the  transit  is  set  on  a  ter- 
restrial mark,  the  altitude  of  the  sun,  the  angle  mark- 
station-sun,  and  the  times  are  taken  and  recorded.  The  angle 


TRANSIT    SURVEYING. 


57 


PZS  is  calculated  and  the  azimuth  of  the  line  from  station  to 
mark  can  be  found  by  addition  or  subtraction. 

70.  Meridian  Without  Calculation. — If  the  meridian  is  to 
be  located  directly  by 
observation,  some  so- 
lar attachment  like 
that  of  Fig.  33  is  nec- 
essary. To  locate  a 
meridian  by  this 
method  we  proceed  as 
follows  : 

1.  Make  the  usual 
five  adjustments     for 
the  transit,  three  for 
the   ordinary     transit 
and  two  for  the  solar 
attachment. 

2.  Bring  the     line 
of  sight  of  solar  tele- 
scope into  the  vertical 
plane    of   the    line    of 
sight  of     the     transit 
telescope. 

3.  If  declination  of 
sun  is  south    (north) 
depress   (elevate)  the 
transit     telescope     an 
amount   equal   to   the 
declination     corrected 
for     refraction,     then 
bring   the    solar   tele- 
scope     to      a      hori- 


Fig. 


zontal  position  by  means  of  its  bubble  tube.  The  lines  of  sights 
of  the  telescope  will  now  include  an  angle  equal  to  the  cor- 
rected declination. 

4.     Elevate   the   transit   telescope   till   the   vertical    arc  reads 
the  co-latitude  of  the  place. 


58  SURVEYOR'S    HAND    BOOK. 

5.  Revolve  both  telescopes  on  their  vertical  axes  till  the 
image  of  the  sun  is  bisected  by  the  vertical  wire  of  the  solar 
telescope.  When  this  bisection  is  secured  the  line  of  sight  of 
the  transit  telescope  will  he  in  the  plane  of  the  meridian  and 
will  locate  it. 

71.  Example:  —  On  April  15th,  1907,  the  following  observa- 
tions were  made  on  the  sun  at  the  magnetic  sta-tion,  Austin, 
Texas  (latitude  30°  17',  longitude  !>7°  -44'  02")  : 

Disc  of  Sun.  W.  U.  Time.      Mark,  Station,  Sun—  Angle. 

Right  ...........       Oh.  50m.  57s.  75°       8' 

Left  .......  .....     lOh.    Om.  03s.  75°    40' 

Average  .........     lOh.    Om.    Os.  75°     24' 

W.  U.  Time  (00  meridian)  =  lOh  Om  Os 
.Correction  —  30     56 

Local    Mean    Time   —  Oh  20m  04s 

Time  from  Greenwich  mean  noon  to  Austin. 
Mean  noon  —  Oh  3<»m  :><>s. 

Time  interval  from  Greenwich  noon  to  ohs."  —  4h. 
Declination  at  Greenwich  mean  noon  =  (J°  27'  i"  .00  N. 
Hourly  increase  =  53"  .0(1. 
Total  increase  =  3'  35"  .84. 

Declination  at  time  of  observation  —  0°  30'  38"  .7-1,  " 
Equation  of  time  at  Greenwich,  mean  noon  —  Orn  17.15:=. 
Hourly  decrease  —  <U>2i  is. 
Total  decrease  =  2.504s. 
E.  T.  at  time  of  ohs.  =  Om  14  .Cos. 
Apparent  time  of  ohs.  •=  Oh  28m  40.35s  =  0.480375h. 
f  =  hour  angle  57>Z  =  2.510025h  =  37°  47'  40". 
tan  d 


Log  tan  d  =  9.224108 
log  cos   t  =  0.807745 

log  tan  N  —  0.326363 

N  =  ll°  58'  12 
/  —  N  =  18°  18'  48" 


TRANSIT    SURVEYING. 

cos  N  tan  t 


tan  a—sn- 

/«•/ «  cos  X  =  9.990453 

log   tan   t—   9.889594 

co-log  sin    (I  —  N)  =     .502775. 


log   tan    (7  =  10.382822 
a—  07° -30'  8" 

Azimuth  of  sun  =  292°  29'  52" 
Azimuth  of  mark  =  75°  24'  — (.7°  30'  8" 

=  7°  53'  52" 

72.     Example: — The  following  data  were   taken   at  a  sta- 
tion where  latitude  =  29°  8'  .1  and  longitude  =  97°  23'  W. 


No.     Sun.      Alt.  of  Sun.       M  -b-    s?  & 

Sun 

W.  U.  Time 
=  90  M. 

1         T) 

31° 

53' 

48" 

5° 

58' 

24" 

8h. 

53 

.Om.    A.M. 

2         "(J|         32° 
3         JO         32°. 

22' 
26' 

00" 
12" 

5° 
4° 

33' 
05' 

12" 
24" 

8h. 
8h. 

55.5m.       " 
58.0m.       " 

4         JO 
Mean 

32° 

55' 

30" 

3° 

30'' 

12" 

9h. 

00.5m.       " 

32° 

24' 

22.8" 

4° 

48' 

33" 

8h. 

56 

.75m.    " 

Declination  of 

sun 

at  Greenwich,  me 

an  noon  = 

2m 

.  19.3s. 

Hourly  increase  —  58".4. 

Time  interval  from  G.  noon  to  Observation  =  2h  56.8m. 

Total  increase  in  Declination  —  2'.9. 

Declination  at  time  of  Observation  =  2°  17'. 2s. 

Observed  altitude  of  Sun  =  32°  24'.3S. 

Correction  for  refraction  and  parallax  = — 1'.3. 

True  altitude  of  Sun  =  32°  23'. 1. 

In  the  PZS  triangle  we  have, 

PZ  — Go3  51'  54"  =  co-lat. 

.P5  — 92°  17'  r2"=:co-deCo 

ZS  —  $r  36'  54"  =  co-alt 
/.  2^  =  210°  46'  00". 
^  —  105°  23' 

.-=iy  5'  48" 


60  SURVEYOR'S    HAND    BOOK. 

sin  5         sin  (s — codec) 

(Cos  i  pzsy- '  --  sin  co_alt  sin  co_lat 

Log  sin  s  =    9.084155 
Loi  sin  (s-codec)  =    9.355249 
cologsin  co-lat  =    0.073417 
colog  sin  co-lot  =    0.058749 
2  fog  o>s  iPZS  =  19.471570 
log  cos  I PZS  •=    9.735785. 
.'.     \PZS=--    57°     1'     40" 
PZS  '=  114°     3'     20" 

Azimuth  of  sun  at  time  of  obs.  =  294°    3'     20" 
Angle  Mk-Sta-Sun  =  .  4°  48'     33" 

Azimuth  of  Mark  =  298°  51'     53" 

Bibliography. — "Theory  and  Practice  of  Surveying."  By 
J.  B.  Johnson.  This  is  one  of  the  best,  most  practical,  and  com- 
prehensive books  upon  higher  surveying.  It  includes  a  discus- 
sion of  the  engineering  instruments  in  their  use  in  ordinary  and 
higher  surveying,  leveling,  topographic,  hydrograohic,  railroad, 
and  earthwork  surveying. 

"The  Principles  and  Practice  of  Surveying "  By  Breed  and 
Hosmer.  520  pages.  This  is  a  rather  full  treatment  on  the  use, 
care,  and  adjustments  of  instruments,  land  surveying,  traverse 
lines,  meridians  and  latitude,  city  surveying,  mine  surveying, 
plotting,  specimen  note  books  and  computations. 

"Plane  Surveying."  By  Wm.  G.  Raymond.  485  pages.  This 
is  a  full  discussion  of  the  construction  and  use  of  the  engineer- 
ing field  instruments,  methods  of  land,  city,  hydrographic,  etc., 
surveying,  and  an  ample  treatment  of  the  slide  rule  (an  un- 
usual feature  of  a  work  en  surveying),  and  an  excellent  set  of 
tables. 

"Surveying  Manual."  By  W.  D.  Pence  and  Milo  S.  Ketchum. 
252  pages.  This  is  one  of  the  most  valuable  hand-books  or  field 
manuals  now  in  print.  While  it  is  modest  in  size,  it  covers  in  a 
satisfactory  way  the  usual  problems  confronting  the  surveyor 
and  engineer.  A  distinguishing  feature  is  the  sample  pages  of 
note  books  executed  in  freehand  lettering. 


CHAPTER  IV. 
CALCULATION  OF  AREAS. 

73.  Latitude  and  Departure  of  a  Course.— Given  a  course 
AB,  Fig.  34,  and  a  meridian  through  one  end  of  the  course,  and 
a  perpendicular  B2  from  the  other  end  upon  the  meridian. 
Then  A -2  is  called  the  latitude  of  the  course,  and  2-B  the  de- 
parture. The  latitude  of  BC  is  B-6  or  3-2.  All  the  latitudes 
that  go  north  are  called  plus  and  all  those  that  go  south  are 
called  minus.  Thus  in  the  figure  the  latitudes  of  AB  and  DA 
are  plus,  while  those  of  BC  and 


CD  are  minus. 
plus  latitudes 


The  sum  of  the 


The  sum  of  the  minus  latitudes 

B6  +  W  =  2-4. 

The  algebraic  sum  of  all  the  lati- 
tudes is  equal  to  zero. 

All  east  departures  are  plus  and 
all  west  departures  are  minus. 
Thus  the  departure  of  AB  and  BC 
are  plus,  while  the  departures  of 
CD  and  DA  are  minus,  The  sum 
of  the  plus  departures  2B  +  6C  — 
3C,  while  the  sum  of  the  minus  or 
west  departures  5C  +  D4  =  C3.  Fig.  34. 

The  algebraic  sum  of  all  the  departures  is  equal  to  zero. 

In    the  triangle  A2B,  let  the  length  AB  =  1,  and  the  angle    BA2 
==  B  (called  the  "bearing*'). 
But  A'2  =  AB  cosine  BA2,  that  is, 
Latitude  ^  length  X  cosine  of  bearing.. 

.  •  .     L  —  I  cosine  B  .............  ....  ____  ,..*....  ____  (7) 

Also,  B2  =  AB  x  sine  of.BA2,  that  is, 

Departure  =  length  X  sine  of  bearing, 

.  •  .    D^l  sim  B  „  .  .  .  .  .  .  .  .  ,  .  ,„  .  .  .  .  .  0  .  „  .  .  .  ,  .  .  .  .  .  .  .  .  .(8) 

61 


SURVEYOR'S    HAND    BOOK. 

Squaring  7  and  8,  and  adding,  we  get, 

L2  +  £>2  =  P  (Cos*B  +  Stn2B). 

But  Sm2B  +  Cos2B  =  1,  _ 

.  •  .     L2  +  D*  =  P     .  '  .     Z  =  VLT+~D*  ...............  (9) 

Dividing  8  by  7,  we  get, 

Departure 
Tangent  5=  .  ...........................  (10) 


Example:  —  The  field  notes  of  a  farm  are  given  in  the  fol- 
lowing table: 

Course.  Bearing.  Distance. 

AB  .....................  N27°37'E  48.6  chains 

BC.  ....................  S67°14'E  6->.4  chains 

CD  .....................  S38°2S'W  52.0  chains 

D/f  .....................  N65°15'W  55.0  chains 

To  find  the  latitudes  and  the  departures  it  is  convenient  to 
proceed  by  finding  the  natural  sines  and  cosines  of  all  the  bear- 
ings, and  arranging  them  under  the  latitudes  and  departures  as 
follows  : 

Latitudes.  Departures. 

Cosine.    Distance.    Latitudes.  Sine.      Distance.    Departures. 

.88674          48.6  43.10  .46226  IH.-i  22.47 

.38698          65.4          —25,31  .9220!)          65,1  00.30 

.78297          :>2.<i          —41.18  .02206          52.0          —32.72 

.41866          55.0  2:!.o:5  .90814  55.0  —49.1)5 

The  latitudes  are  found  by  multiplying  the  cosine  by  the  dis- 
tance, and  the  departures  by  multiplying  the  sine  by  the  dis- 
tance. 

74.  Traverse  Tables.  —  To  facilitate  calculation  in  the  of- 
fice, tables  have  been  prepared  by  which  the  latitude  and  de- 
parture can  be  obtained  without  arithmetical  calculation.  Thus 
for  any  angle  under  45°  and  for  all  distances  from  1  to  loo  tlrj 
latitude  and  departures  are  calculated  and  tabulated.  Thus  for 
an  angle  of  10°  we  find  : 

Sin  10°  =  .17365 

Cos  10°  =  .98481 
Then  for  any  distance  x  we  have 

Departure  =  .17365  .r; 

Latitude  =  .98481  x. 


CALCULATION    OF    AREAS. 


63 


Dist. 
1 

Lat. 

9848 

2 

1  97 

3    

295 

4    

,5.94 

4  02 

6 

.  ,      .       5  91 

(i89 

8    . 

7.88 

9    

8.86 

10    . 

..9.85 

11  Deg.. 

12  Deg. 

Lat.      Dep. 

Lat. 

Dep. 

.98          .19 

.98 

.21 

1.96          .38 

1.96 

.42 

2.94          .57 

2.93 

.62 

3.93           76 

3.91 

.83 

4.91          .95 

4.89 

1.04 

5.89 

.14 

5.87 

1.25 

6.87        ] 

.34 

6.85 

1.46 

7.85        1 

.53 

7.83 

1.66 

8.83 

.72 

8.80 

1.87 

9.82        1 

.91 

9.78 

2.08 

Now,  if  we  give  to  ,\-  values  from  1  to  10,  the  following  results: 

10  Deg. 

Dep. 

.17 

.35 

.52 

.69 

.87 

1.04 

1.22 

1.39 

156 

1.74 

In  the  same  way  the  latitudes  and  departures  can  be  calcu- 
lated for  all  distance  desired  and  for  angles  as  minute  as  space 
will  allow.  Some  works  on  surveying  have  traverse  tables  for 
all  distances  from  1  up  to  100  and  for  all  angles  15'  apart  from 
zero  to  90°. 

75.  Example: — If   the  distance  is  56.8  chains  and  the  bear- 
ing is  N.  10°  E.  we  divide  up  the  number  into  50,  6,  and  .8  and 
find  the  latitude  and  departure  of  each   separately  and  add  the 
results.     We  look  for  the  latitude  and  departure  of  5  and  multi- 
ply the  result  by  10  to  get  the  lat.  and  dep.  for  50.     If  bearing  is 
10°,  we  have  for  5  chains, 

Lat.  ==  4.92 
Dep.  ==•  .87 
Hence,  we  have, 

For  50,  lat.  =  4.92   : 

6,  lat.  =  5.91 
"      .8,  lat.  =    .788 

Lat.  for  56.8  chains  . 

Dep.  for  50  =  .87     > 

6  = 

Total  dep.  for  5(5.8  chains =    9.88 

.As  an  exercise,  find  the  latitude  and  departure  for  bearing  of 
12°  and  a  distance  of  37.48  chains. 

76.  Error  of  Closure. — In  surveying  parties  the  surveyor 
is  usually  the  only  skilled  man  in  the  party.    The  chainmen  are 


10 


64  SURVEYOR'S    HAND    BOOK. 

usually  picked  up  in  the  locality  and  are  not  supposed  to  be 
trained  in  this  work.  It  is  assumed  in  balancing  the  survey  that 
the  errors  are  due  to  the  chaining  and  that  the  surveyor  reads 
the  bearings  correctly.  Tf  in  balancing  the  error  is  greater  than 
1  in  500  the  farm  should  be  resurveyed.  The  error  in  latitude 
or  departure  is  the  amount  that  the  algebraic  sum  of  the  lati- 
tudes or  departures  lacks  of  being  zero.  The  error  of  closure  is 
found  by  squaring  the  error  in  latitude  and  the  error  in  de- 
parture and  taking  the  square  root  of  their  sum  and  dividing 
this  result  by  the  perimeter  of  the  farm.  This  is  simply  divid- 
ing the  distance  you  miss  the  beginning  corner  by  the  length  of 
the  perimeter  of  the  farm. 

Find  the  latitudes  and  departures  for  the  following  courses  : 


Course.          Bearing.  '      Latitudes-     Departures. 


AB  N23°30'E  '255.72  234.49  101.96 

BC  N68°E  182.1  68.22  168.84 

CD  S3°47'W  329.42  —328.67  —21.74 

DA  N84°15'W  249.92  25.04  —248.66 

Thus  in  the  example  the  error  in  latitude  is  —  .92  and  the 
error  in  departure  is  +  .40,  that  is,  we  went  nonh  1527.  To  and 
south  +  328.67,  which  leaves  us  +  .92  south  of  A.  We  went 
east  270.80  and  west  270.40,  which  leaves  us  +  .40  west  of  A 
at  some  point  A'. 

But  A  A'  =  v/(.92)2  +  (.40)2  • 

V(.92)g  +    (.4Q)2  1 

And  the  error  of  closure  =  1Q17  ^  =  JoJI 

77.  Balancing  a  Survey.  —  Theoretically  the  algebraic  sum 
of  the  latitudes  is  equal  to  zero,  and  the  same  is  true  of  the  de- 
partures. But  in  actual  survey  work  these  sums  never  are  equal 
to  zero,  owing  to  unavoidable  errors.  These  errors  must  be 
distributed  in  proportion  to  the  length  of  the  courses.  We  see 
that  the  error  in  departure  is  .40,  which  must  be  distributed 
among  the  courses  in  proportion  to  their  lengths. 

The  total  distance  around  the  farm  (the  perimeter)  is 
1017.16  varas,  and  the  total  error  in  departures  is  .40  and  that 


CALCULATION    OF    AREAS. 


65 


for   latitudes   is   .92.     The  error   of  any  course  is  to   the  total 
•error  as  the  length  of  any  course  is  to  the  perimeter. 

If  the  compass  was  used  in  making  the  survey  this  rule  for 
balancing  should  be  followed  even  if  some  of  the  courses  are 
due  north-south,  or  due  east-west.  The  compass  cannot  define 
the  angle  accurately  and  there  is  as  much  probability  of  error  in 
angle  in  a  due  north  course  as  there  is  in  a  course  whose  bearing 
is  N.  26°  E.  Again,  in  some  of  the  older  states  the  magnetic 
bearings  are  read  and  a  course  that  is  north  at  the  present  time 
could  make  one  degree  with  the  magnetic  meridian  twenty  years 
hence.  If  the  practice  of  distributing  the  errors  in  departure 
(or  latitude)  among  those  courses  that  have  departure  be  fol- 
lowed in  the  calculation  of  the  first  survey,  the  above  method 
would  have  to  be  followed  in  the  last  survey.  Thus  the  same 
surveyor  would  get  different  results  for  the  area  of  the  farm. 
The  usual  rule  should  be  followed  in  all  cases  for  a  compass 
survey. 

Therefore,  the  error  for  any  course  = 
total  error 
perimeter    X  len^th  of  course' 


Con 
AB 

BC 
CD 
DA 

'ections  for 
.92 

Latitude  of 

.23 
.16 
.30 
.23 

Cor 
AB 

BC 
CD 
DA 

rections  for 
.40 

Departure  o 

f 
.10 

.07 
.13 

.10 

.40 

~  1017.16 
.92 

X  255.72  — 

"  1017.16 
.40 

X  255.72  — 
x;  181.1    = 
X  329.42  = 

"  1017.16 
.92 

X  182.1    — 

~  1017.16 
.40 

=  1017.16 
.92 

X  320.42  — 

~  1017.16 
.40 

"  1017.16 
Total  for 

X  249.92  — 

1017.16  A  ^y-^_ 
Total  for  Departure  = 

Latitude  = 

.92 

fhese  are  arranged  in  the  following  table : 

Corrections. 
Lat.  Dep. 


Course 


Cor.  Lat. 


AB 
BC.. 
CD. 
DA, 


.23 
.16 
.30 
.23 


.10 
.07 
.13 
.10 


234.72 
68.38 

-328.37 
25.27 


Cor.  Dep. 

101.86 
168.77 

—21.87 
—248.76 


SURVEYOR'S    HAND    BOOK. 


The  sum  of  the  uncorrected  plus  latitudes  b  327.75,  and 
that  of  the  minus  latitudes  is  328.67;  all  the  plus  latitudes  must" 
be  increased  by  their  corrections,  and  the  minus  must  be  de- 
creased by  their  corrections.  If  these  corrections  are  applied 
property  we  will  get  the  numbers  in  the  column  "Cor.  Lat.," 
which  means  corrected  latitudes.  The  sum  of  the  plus  depart- 
ures is  270.80  and  the  sum  of  the  minus  departures  is  270.40;  the 
sum  of  the  plus  departures  is  greater  by  .40;  therefore  the  minus 
departures  must  be  increased  and  the  plus  departures  decreased. 
The  column  headed  "Cor.  Dep.''  gives  the  corrected  departures, 
78.  The  Double  Meridian  Distance.  -  -  The  reference 

meridian  generally  passes  through 
the  most  westerly  corner  of  the 
land.  The  perpendicular  from  the 
mid  point  of  the  course  upon  this 
meridian  is  called  the  meridian 
distance.  The  meridian  distance 
of  MNj  Fig.  35,  is  xy  where  x 
is  the  midpoint  of  MX.  But  if 
M3,  Ar4  and  O5  are  perpendicular 
to  the  meridian,  M3  +  JV4  =  2*3', 
kO  or  double  the  meridian  distance, 
and  is  called  the  D.  M.  D.  That 
is,  the  DMD  of  any  course  is 
equal  to  the  sum  of  the  two  per- 
pendiculars from  its  ends  upon 


Fig.   35. 


the  reference  meridian. 

The  DMD  of  AT0  =  A'4  +  O5 

=  A'4  +  Ol  +  N6  +  M3 
=  (N4  +  M3)  +  N6  +  06 

That  is,  the  DMD  of  any  course  is  equal  to  the  DMD  of  the 
preceding  course,  plus*  the  departure  of  the  preceding  course 
plus  the  departure  of  the  course  itself.  The  DMD's  of  the  first 
and  last  courses  are  always  equal  to  their  own  departures. 

A  sketch  of  the  farm  whose  latitudes  and  departures  were 
balanced  in  Art.  77  shows  that  A  is  the  most  westerly  corner, 
and  it  will  be  convenient  to  take  our  reference  meridian  through 


"CALCULATION    OF    AREAS. 


this  corner.  Then  the  Double  Meridian  Distance  of  AB  =  De- 
parture of  AB  =  101.86. 

The  D.  M.  D.  of  BC=  101.86  +  101  86  +  168.77  =  372.49. 

The  D.  M.  D.  of  CD  =  372.49  +  168.77  —  21.87  =  51f.39. 

The  D.  M.  D.  of  £>,4  =  519.39  —  21,87  —  248.70  =  248.76. 

The  last  result  proves  the  correctness  of  our  arithmetical 
work,  as  the  DMD  of  the  last  course  should  equal  the  de- 
parture of  that  course. 

If  the  course  AB  does  not  happen  to  be  in  the  first  line  of 
the  table  of  notes,  the  DMD's  can  be  calculated  with  reference 
to  the  most  westerly  comer  without  rearranging  the  table. 

79.    Area   of    a    Farm. — If     we 

M 

drop  perpendiculars  from  the  ends  of 
the  courses  upon  the  meridian  NS. 
Fig.  36,  we  form  trapezoids,  or  trian- 
gles. If  we  survey  around  the  farm 
clockwise,  all  the  areas  determined 
by  the  courses  and  perpendiculars 
that  have  plus  latitudes  will  be  out- 
side the  farm,  while  those  that  have 
minus  latitudes  will  include  part  of 
the  farm  and  part  of  the  area  be- 
tween the  farm  and  the  reference 
meridian.  The  algebraic  sum  of 
the  "minus  areas"  and  "plus  areas" 
is  equal  to  the  area  of  the  farm. 

The  double  area  of  AB2  =  A'2  X  B'2  —  Lat.  XDMD. 

Double  area  2#  C6—2  3  X  (B'l+  C'3)  =  Lat.  X  DMD. 

Double  area  BC  D4  =  3  4  X  (C3  +  D4)  —Lat.  X  DMD. 

Double  area  W  £5=5  4(774  +  £5)  =  Lat.  X  DMD. 

Double  area  »EA=5A    (5E\=Lat.  X  DMD. 

The  areas  of  253C  and  3CD4  have  minus  latitudes  (2  3  and 
3  4)  and  these  areas  are  therefore  called  "minus  areas,"  and 
they  not  only  include  the  whole  farm  but  also  the  areas  between 
the  farm  and  the  reference  meridian.  The  areas  AB2,  4D.E5. 
and  SEA  have  plus  latitudes,  and  are  called  "plus  areas."  If 


Fig.   36. 


68 


SURVEYOR'S    HAND    BOOK. 


we  add  the  "plus  areas"  to  the  "minus  areas,"  there  is  left  the 
area  of  the  farm.  ABODE. 


Co.r« 

Be.,,-,, 

D.sr 

Cor  D«p 

DMD 

Plus    AK.O 

Minu*  Area 

AB 

N38E. 

26ch 

2049 

1601 

20.40 

16.15 

1  6.15 

-3294&00 

BC 

S42E 

34c», 

-25.27 

2275 

-2339 

22.93 

3523 

1402.2637 

CD 

S5IW 

20«t, 

-259 

-15.54 

-1266 

-•3.*3 

62  73 

794.1618 

DA 

N72W 

25«h 

775 

-2378 

765 

•2365 

2365 

1809285 

28  22 

-39  32 

510  3825 

2I904AIA 

-278G 

_  38  76 

5)0.3825 

36 

-      -36 

Z   X  Area   = 

I«S6.0690 

Art 

a  m  S<»  Ch   « 

6430343 

Art 

to  m  Acres  = 

8430JT 

Err 

oca. 

La 

t     Cor 

Dep 

Cor 

C,-  - 

3&*2< 

;  =  .O9 

C,  «  t? 

*«26* 

14 

c,  • 

•     x  34» 

J8 

C.- 

••     X2 

5»  07 

c,. 

1     »  20  = 

ji 

cv 

••     X2 

3'  08 

c*« 

"    »  25- 

.13 

Total 

=    36 

Tetol 

- 

50 

Fig.    37. 

80.  Area  Table.— Placing  the  1)  M  D's  in  the  table  and 
multiplying  each  by  its  corresponding  latitude,  we  find  the  areas 
as  given  in  the  following  table.  Dividing  the  area  in  square 
yards  by  4840  gives  the  area  in  acres : 


Course. 

Bearing. 

Dist, 

Lai, 

Corrections. 

Dep. 

Lat. 

Dep 

AB    . 
BC    .... 
CD   .... 
DA   .... 

N  23  30  E 
N  68        E 
S   3    47  W 
N84  15  W 

255.72 
182.1 
329.42 
249.92 

234.40 
68  22 
—328.67 

25.04 

101.96 
168.84 
—21.74 
—  248.66 

.23 
.16 
.30 
.23 

.10 
.07 
.13 
.10 

CALCULATION    OF    AREAS. 


Cor.  Lat.  Cor.  Dep.  D.  M.  D. 

234.72  101.86  101.80 

68.38  1(58.77  372.41) 

-328.37  —21.87  519.39 

25.27  —248.76  248.76 


Plus  Areas.  Minus  Areas. 

23,905.5792  

28,479.0662  

170,552.0943 

6,286.5652 


55,671.2106 


170,552.0943 
55,671.5652 


Double  area 
.'.  Area  =  57,440.441 95  sq.  yds.  =  23.7357  acres. 


=  114,880.8837 

The    standard    form    of    calculation   of    errors    and    areas    is 
shown  in  Fig.  37. 

PROBLEM  25. — William  James  Farm. 

Course.                                   Bearing.  Distance. 

AB S84°45'E  19.73  chains 

BC S21°E  15.85  chains 

CD S78°W  19.53  chains 

DA N16°W  21.51  chains 

Area  =  35.01575  acres. 

PROBLEM  26.— Cambria  Farm. 

Course.                                   Bearing.  Distance. 

AB S  41  E  100  poles 

BC..                    S  29  W  41  poles 

CD N  69  W  99  poles 

DA N  31  30  E  90  poles 

Area  =  39.357  acres. 

PROBLEM  27. — Oran  Farm. 

Course.                                   Bearing.  Distance. 

AB N5°16'E  205  6  varas 

BC N28°30'W  263,5  varas 

CD S16°E  255.6  varas 

DE S3°15'E  210.6  varas 

EA N84°15  W  227.7  varas 

Area  =11. 958  acres. 

PROBLEM  28. — Diego  Blanco  Farm. 

Course.                                     Bearing.  Distance. 

AB N56°E  8.18  chains 

BC S16°E  5.39  chains 

CD . .               S59°30'W  7.49  chains 

DA N17°10'W  4.87  chains 

Area— 3.97879  acres. 


70 


SURVEYOR'S    HAND    BOOK. 


PROBLEM  29.— Bowie  Blanca  Farm. 
Course.  Bearing. 

AB N56°E 

BC S1(>°E 

CD S59°30'W   ' 

DA N17°10'W 

•  Area  =  3.9786  acres. 

PROBLEM  30. — Bantello  Farm. 
Course.  Bearing. 

AB N33°E 

BC S67°E 

CD S38°40'W 

DA N48°W 

Area  =  27.5937  acres. 

PROBLEM  31. — Leon  Brooks  Farm. 
Course.  Bearing. 

AB N5°16'E 

BC S17°10'E 

CD S56°30'E 

DA N85°15'W 

Area  =  2.80  acres. 

PROBLEM  32. — Francis  Estell  Farm. 
Course.  Bearing. 

AB N5°E 

BC S70°15'E 

CD S16°E 

DA S56°W 

Area= acres. 

PROBLEM  33.— Jnnn  Viego  Farm. 
Course.  Bearing. 

AB N59°30'E 

BC N78°E 

CD S23°30'W 

DA N56°30'W 

Area  = acres. 

PROBLEM  34. — John  Bruce  Farm. 
Course.  Bearing. 

AB..  ..N87°E 

BC S59°17'W 

CD S84°45'W 

DA N16°W 

Area  =  .  .       .  acres. 


Distance. 
540.0  feet 
356.0  feet 
524.0  feet 
321.2  feet 


Distance. 
14  chains 

18  chains 

19  chains 
16  chains 


Distance. 
205.6  varas 

115.6  varas 

207.7  varas 
227.3  varas 


Distance. 
750.0  feet 
300.0  feet 
356.8  feet 
540.0  feet 


Distance. 
7.94  chains 
4.88  chains 
10.77  chains 
8.74  chains 


Distance. 
376.0  varas 
260.0  varas 

117.3  varas 

128.4  varas 


CALCULATION    OF    AREAS. 


71 


PROBLEM  35.— H.  Yandell  Farm. 
Course.  Bearing. 

AB N59°17'E 

BC S9°15'W 

CD S68°W 

DA N21°W 

Area^ acres. 

PROBLEM  30. — Sidney  Dean  Farm. 
Course.  Bearing. 

AB N67°W 

BC S59°03'W 

CD S49°E 

DA N47°30'E 

Area  =  23. 53 1  acres. 

PROBLEM  38. — George  Pierce  Farm. 
Course.  Bearing. 

AB N26°45'E 

BC N11°42'E 

CD N40°05'E 

DE S41°42'E 

EF S40°15'W 

FG S20°15'W 

GA N69°34'W 

PROBLEM  38.— Old  Perry  Farm. 
Course.  Bearing. 

AB • S73°45'E 

BC S53°30'E 

CD S47°12'W 

DE '.  ...S55°00'E 

EF..... S48°00'W 

FG....  N63°57'W 

GH N53°25'W 

HA N3°30'W 

Area  =  48.15785  acres. 

PROBLEM  39. 
Course.  Bearing. 

AB N57°42'E 

BC S30°E 

CD ..-    S38°40'W 

DE S23°W 

EF S60°E 

FA N30°57'W 

Area  =  34.2404  acres. 


Distance. 

260.0  varas 
151.6  varas 

182.1  varas' 
94.3  varas 


Distance. 
11.52  chains 
18.30  chains 
14.30  chains 
21.17  chains 


Distance. 

60  feet 

364  feet 

1038  feet 

1650  feet 

341  feet 

322  feet 

1640  feet 

Distance. 

21.60  chains 

22.72  chains 

6.95  chains 

8.38  chains 

5.55  chains 

3.12  chains 

41.60  chains 

6.36  chains 


Distance. 

17.47  chains 

18.47  chains 

2.27  chains 

1.88  chains 

13.05  chains 

20.21  chains 


72  SURVEYOR'S'  HAND    BOOK. 

PROBLEM  40. 
Course.  Bearing.  Distance. 

AB N3°53'E  7.70  chains 

BC S82°8'E  39.05  chains 

CD S83°42'E  14.39  chains 

DE S56°9'W  14.26  chains 

EA N80°3'W  42.30  chains 

Area  =  40.604  acres. 

PROBLEM  41. 
Course.  Bearing.  Distance. 

AB N60°05'E  19.90  chains 

BC S13C32'W  9.03  chains 

CD S27°20'W  9.77  chains 

DE S43°40'W  5.«7  chains 

EA N30°43'W  13.24  chains 

Area  =  16.3432  acres. 

81.  Courses  of  No  Latitude  or  Departure. — If  a  survey 
is  made  with  the  transit,  the  sum  of  the  interior  angles  of  the 
polygon  should  equal  two  right  angles  taken  as  many  times  as 
the  polygon  has  sides  less  two.  The  error  should  not  amount  to 
more  than  three  minutes,  unless  the  number  of  sides  is  large. 
In  a  transit  survey  there  can  be  very  little  error  in  the  angular 
measurements  and  all  errors  in  latitude  and  departure  are 
largely  due  to  errors  in  chaining.  If  a  transit  line  is  due 
north  it  is  presumed  that  it  is  in  the  true  meridian  and  there- 
fore has  no  departure.  Similarly  if  the  course  is  due  east  it  has 
no  latitude,  and  if  the  angles  check  within  three  minutes  (3'), 
the  errors  must  be  distributed  on  the  assumption  that  they  were 
due  10  the  chaining.  The  practice  is  to  distribute  the  errors  in 
latitude  (departure)  among  those  courses  that  have  latitude  or 
departure.  Thus  no  north-south  course  would  receive  a  correc- 
tion for  departure  as  its  original  departure  and  also  its  balanced 
departure  is  zero.  Similarly  a  due  east-west  course  receives  no 
correction  for  latitude.  Hence  if  a  course  is  north  (east)  its 
length  is  omitted  in  the  perimeter  of  the  field  in  calculating 
the  errors  in  departure  (latitude).  The  following  rules  are 
used  in  balancing : 

Rule  No.  1. — Distribute  all  errors  in  latitude  (departure)  in 
proportion  to  the  length  of  the  courses.  If  any  course  is  north 


CALCULATION    OF    AREAS. 


73 


(east)  its  length  is  omitted  from  the  perimeter  of  the  Held. 
Error  in  latitude  (departure)  for  any  course  is  to  the  whole 
error  in  latitude  (departure)  as  each  course  is  to  the  corrected 
perimeter. 

Rule  No.  2. — The  error  in  latitude  (departure)  in  any  course 
is  to  the  whole  error  in  latitude  as  the  latitude  of  the  course  is 
to  the  sum  of  all  the  latitudes. 

The  transit  is  rapidly  becoming  the  surveyor's  instrument,  as 
there  is  greater  demand  for  accuracy  with  the  advanced  price 
of  land.  The  needle  is  inaccurate  at  best  and  when  we  consider 
the  effect  of  barbed  wire  fences,  telephone  and  telegraph  wires, 
local  attraction  and  other  similar  influences  that  render  the 
needle  unstable,  its  efficiency  as  an  instrument  of  precision  is 
rendered  doubtful  in  the  extreme. 

Rule  No.  2  is  by  far  the  most  logical  in  transit  surveys  and 
should  be  used  in  balancing,  and  it  has  the  advantage  that  it  is 
automatic  in  that  it  finds  no  error  in  departure  for  north-south 
courses  or  in  latitude  for  east-west  courses. 

82.  Example: — In  the  following  survey  the  errors  were 
distributed  in  proportion  to  the  length  of  those  courses  that 
have  latitude  or  departure : 


Course 
AB 
BC 
CD 
DE 
EA 

Bearing. 

N47°E 
East 
S32°E 
South 

N77°30'W 

D.  M.  D. 

2966 

Dist.        Lat.       Dep. 
40.57       27.67       29.67 
34.59       00.            34.59 
27.10  —22.98        14.36 
22.01  —22.01           .00 
80.51       17.43  —78.60 

Plus  Area. 
819.8024 

Cor- 
rections.       Cor. 
Lat.    Dep.      Lat. 
.03       .01       27.64 
.00       .00        
.02       .00  —23.00 
.01       .00       22.02 
.05       .01       17.38 

Minus  Area. 

Cor. 
Dep. 
29.66 
34.59 
14.36 

—  is.'e'i 

9391 

142.86 
157  22 

3,285.7800 
3,461.9844 

7861 

13662418 

2186.04-12 

6,747.7644 
2,184.0/42 

Double  area  =  4,561.7202 
Area  =  2,280.8601   sq.  chains  =  228.08601   acres. 


74 


SURVEYOR'S    HAND    BOOK. 


If  the   errors   are   distributed   in  proportion  to  the   latitudes 
and  departures,  the  result  is  as  follows: 

Correc- 
tions.     Cor. 
Lat.  Dep.  Lat. 


.03  .00 
.00  .00 
.03  .00 
.03  .00 
.02  .02 


27.64 

0.00 

—23.01 

—23.04 

17,41 


Cor. 

Dep.      D.  M.  D 

29.67        29.67 

34,59        93.93 

14.36 

0.00 


—78.62 


142.88 
157.24 
78.62 


820.0788 


1368.7742 

2188.8530 


3287.0688 
3465.5696 

6753.2371 

2188.8530 


Double  area  =  4564.3844 

.-.  area  =  2282.1922 

=  228.22  acres 


Distance. 
20.0  chains 
8.0  chains 
28.0  chains 
23.3  chains 


Distance. 

10  chains 

11  chains 
17  chains 
20  chains 


PROBLEM  42. 
Course.  Bearing. 

AB N36°9'E 

BC East 

CD South 

DA N59°2'W 

Area  =  34,3779  acres. 

PROBLEM  43. 
Course.  Bearing. 

AB N39°30'E 

BC East 

CD South 

DA N61°W 

Area  =  19.158  acres. 

PROBLEM  44. — Find  the  area  of  the  following:  Beginning  at 
a  stake  in  road  762.5  feet  west  from  Chisholm's  southwest  cor- 
ner; thence  N.  0°  30'  E.  661  feet;  thence  up  branch  S.  81  W. 
117  feet,  S.  22  W.  124  feet,  S.  8  W.  87,  S.  70°  30'  W.  162  feet, 
then  S.  27°  30'  W.  153,  S.  31°  30'  E.  62  feet,  S.  34°  "W.  94  feet, 
E.  304  feet,  S.  5°  W.  129  feet  to  middle  of  said  road;  thence  E. 
along  said  road  116  feet  to  beginning. 

PROBLEM  45. 
Course.  Bearing.  Distance. 

AB N39°E  20 

BC East  8 

CD South  28 

DA..  ..N60°W  23 


CALCULATION    OF    AREAS. 


75 


83.  Area  by  Co-ordinates.— If  the  co-ordinates  of  each 
corner  of  the  farm  are  given  with  reference  to  two  axes  OX 
and  OY,  we  can  find  the  area  by  dropping  perpendiculars  from 
each  corner  on  either  axis,  as  OX,  Fig.  38. 


Let  Oa,  Ob,  Oc  and  Od  —  x&,  x^,  xc  and 
A  a,  Bb,  Cc  and  Dd  =  -ya,  yb,  yc  and  y&. 


repectively;  and 


Now 


area  aABb  =•-  ab 


area  bBCc  ==  be 


area  cCDd  =  cd 


area  dDA  a  =  da 


(Aa  +  Bo) 

2 
(Bb  +  Cc) 

2 
(Cc  +  Dd) 


(Dd  +  Aa) 


area  of  farm  =  aABb  +  bBCc  —  cCDd  —  dDAa  =  (#b  —  x&) 
O'a  +  y\>} 


.  •  .  Double  area 

(  Vc—  >'a)  - 

Similarly 
Double  area  = 


(y<i  —  y\>)  +  x\,  (y&—yc)  +  xc  (yb  —  yd)    + 


(xt\  —  x\>)    +   Tb  (*&  —  *e)    + 


This  can  be  crystallized  into  the  following  rule  :  To  find  the 
double  area,  multiply  each  abscissa  (ordinate)  by  the  difference 
of  the  adjacent  ordinates  (absc'issas)  taken  in  order. 


76 


SURVEYOR'S    HAND    BOOK. 


EXAMPLE. — Find  the  area  of  the  farm  whose  co-ordinates  are 
(2,  6),  (6,  10),   (12,  8),  (4,  2). 
Diff.  of 
X's. 
2 
10 
—2 


n 


X. 

2 

6 
12 

4 
Area, 


7. 

6 

10 

8 

9 


—10 


Fig.   39. 


Course. 
AB 
BC 
CD 
DA 


Bearing. 
N31°E 
N33°E 

N36°E 


Diff.  of 

Area.        Y's.       Area. 
12  8  16 

100  2  12 

_16        —8        —96 
_20        —2         —8 

.76.0 76.0 

PROBLEM  46. — Find  the  area 
by  both  methods  of  the  farm 
whose  co-ordinates  are  (2,  4). 
(4.  8),  (12,  12),  (16,  4), 
(10;  0).  Answer  .96. 

PROBLEM  47. — Find  area  of 
polygon  whose  co-ordinates  are 
(0,0),  (0,12),  (10.9),  (18,  14), 
(22,  13),  (9,  0). 

84.  Traversing. — When  it 
is  desired  to  find  the  bearing 
and  distance  of  one  point  from 
another,  a  survey  is  run  from 
the  initial  point  to  the  final, 
making  as  many  straight 
courses  as  desired.  The  lati- 
tudes and  departures  of  these 
courses  are  calculated,  and  the 
closing  course  is  a  lost  course 
whose  bearing  and  length  are 
desired  and  can  be  found  by 
formulas  8  and  9. 

85.  Example. — F  i  n  d  the 
bearing  and  length  of  AD  in 
the  following: 

Distance      Latitude.  Departure. 
20  chains  17.14  10.30 

24  chains  20.13  13.07 

26  chains  21.03  15.28 


CALCULATION    OF    AREAS.  77 

38  65 
The  tangent  of  the  bearing  =  ^^  =  .66295. 

Therefore  the  bearing  =  N  33°  32'  E 

Length  =  v/(58.20)2+  (38.6S)2  =  69.86. 

86.  Approximate  Traversing.  —  Where  the  bearings  of  the 
different  courses  of  a  traverse  do  not  differ  by  more  than  6° 
the  bearing  can  be  found  by  an  application  of  the  57.3  rule.  Let 
ABCD,  Fig.  39,  be  a  traverse,  and  let  the  bearings  be  as  in  the 
preceding  example.  Take  a  reference  line  and  let  a,  b,  and  c 
be  the  angles  that  AB,  EC,  and  CD  make  with  this  line  AG. 

IB         "'' 

=  WJ~3 

HC—^- 

ZO   —   (-7   o 

57.3 

«>-£» 

Let  x  =  angle  that  AD  makes  with  reference  line  AG. 


But  AD  =  AB+BC  +  CD,  nearly 


x 

(li  +  !«+!.)= 


67         i       «.  57.3 

ali  +  bls  +  cl* 

li  +  l'2+la 
ali+blz  +  ck 

57.3 

If  B  =  bearing  of  the  reference  line  and  we  add  B(\i  +  12H-  18) 
to  each  side,  we  get  : 


/n      v^  ._. 

I1  +  l2+la 

That  is,  multiplying  each  bearing  by  its  length,  and  dividing 
the  sum  of  the  results  by  the  sum  of  the  lengths  of  the  courses 
gives  the  bearing  required. 

Let  a  =  32°,  b  ==  33°,  c  =  36°,  AB  =  20,  BC  =  24,  CD  ***  26, 
find  bearing  of  AD. 


78 


SURVEYOR'S    HAND    BOOK. 


PROBLEM  48. — Find  the  approximate  bearing  of  AD  from  the 
following  notes : 

Course.  Bearing.  Distance. 


AB S28E 

BC S32E 

CD S30E 

DA.. 


8 


20  chains 
18  chains 
22  chains 

87.     Irregular  Boundaries. 

— It  often  happens  that  a  creek 
or  river  is  the  boundary  of  a 
tract  of  land  and  the  land  fol- 
lows the  meanders  of  the  river. 
Thus  the  field  notes  of  a  cer- 
tain larm,  Fig.  40,  are  as  fol- 
lows : 

Beginning  at  a  pecan  tree 
marked  X  on  Stone  Creek, 
thence  N.  36°  9'  E.  to  a  stone 
in  the  prairie  29  chains  ;  thence 
E.  8  chains  to  a  cottonwood 
marked  H  on  the  west  bank  of 

FI      4Q  Mill   Creek;   thence    with    the 

meanders  of  Mill  Creek  to  the 

junction  of  Stone  Creek;  thence  up  Stone   Creek   to   the   begin- 
ning. 

The  following  offsets  were  taken : 


CD 

DA 

Dist. 

Offset. 

Area. 

Dist. 

Offset. 

Area. 

00  chains 

00.    chains 

.00 

acres 

00. 

chains 

00.    chains 

.00    acres 

4  chains 

2.0  chains 

.4 

acres 

5. 

chains 

2.3  chains 

.575  acres 

7  chains 

2.5  chains 

.675  acres      9. 

chains 

2.5  chains 

.960  acres 

9  chains 

2.2  chains 

.47 

acres 

14. 

chains 

2.1  chains 

1.15    acres 

12  chains 

1.0  chains 

.48 

acres 

17. 

chains 

1.8  chains 

.32    acres 

15  chains 

1.4  chains 

.36 

acres 

19. 

chains 

1.4  chains 

.07    acres 

20  chains 

1.8  chains 

.80 

acres 

20. 

chains 

.0  chains 

.05    acres 

24  chains 

2.0  chains 

.76 

acres 

21. 

chains 

1.0  chains 

.09    acres 

26  chains 

1.7  chains 

.37 

acres 

22. 

chains 

.8  chains 

.09    acres 

28  chains 

0.0  chains 

.17 

acres 

23.3  chains 

.0  chains 

.052  acres 

4.485  acres 


3.852  acres 


CALCULATION  OF    AREAS.  79 

Area  of  farm  ABCD  =34.3779  acres 

Area  of  offsets  from  C  to  D          =   4.1250  acres 

Area  of  offsets  from  D  to  A  =   3.852    acres 


Total   area  of  farm  with   offsets  =  42.7149  acres 
The  land  lines  run  up  to  the  bank  if  the  stream  is  navigable. 
PROBLEM  49. — The  following  offsets  were  taken  where  R  and 
L  refer  to  right  and  left  of  the  line  being  surveyed.     Find  the 
total  area  of  farm  if  bounded  by  straight  sides  AB  and  BC  and 
the  meanders  of  the  streams  to  which  offsets  were  taken  from 
points  along  CD  and  DA. 

Length  along        Offsets  Length  along  Offsets 

CD  DA 

00  00 

3  .6  R  3  .4  L 

5  .8  R  5  .6  L 

7  .7  R  7  .8  L 

8  .3  R  10  .4  L 

9  0.0  12  0.0 
11                             .3  L                           14  .3  R 
13                             .5  L                          .16  .5  R 
15                             .4  L                           18                             .4  R 
17                          0.0                              20                          0.0 

88.  Discrepancies. — It  often  happens  that  a  survey  is  found 
where  little  care  was  exercised  in  the  original  survey  when  the 
grant  or  patent  was  taken  up.  If  there  are  errors  in  the  field 
notes  of  the  original  grant  and  there  are  no  natural  objects  to 
which  reference  was  made,  it  is  very  difficult,  if  not  impossible, 
to  re-establish  the  old  lines.  But  if  natural  objects  were  re- 
ferred to  in  the  original  field  notes,  and  these  obj.ects  can  be 
found  and  identified,  the  re-establishment  of  the  old  survey  is 
possible  and,  sometimes,  comparatively  easy.  Corners  are  often 
defined  or  witnessed  by  natural  objects,  while  the  distances  in 
the  field  notes  do  not  agree  with  such  witness  objects.  In  such 
cases  the  "natural  objects  control  and  the  corners  must  be  located 
as  called  for  by  the  natural  object  irrespective  of  the  length  of 
the  lines  in  the  notes.  If  a  line  begins  at  a  well  known  tree 
and  runs  with  a  certain  bearing  to  the  middle  of  a  certain 


80  SURVEYOR'S    HAND    BOOK. 

stream,  and  thence  with  the  meanders  of  the  same,  etc.,  the  line 
must  go  to  the  center  of  the  stream,  although  the  distance  of 
the  line  may  fall  short  or  exceed  that  called  for  in  the  recorded 
field  notes. 

PROBLEM  50. — The  area  was  calculated  to  he  39.357  acres. 
Find  the  area  of  the  farm  if  the  line  DA  was  a  random  line  from 
which  offsets  were  taken  to  a  small  creek  on  the  left  of  DA,  and 
completely  outside  the  farm  as  given  in  problem  20.  The  follow- 
ing are  the  field  notes  for  the  offsets  taken  along  DA  : 
Dist.  from  D  Offsets  to  left 

0  0 

16  8 

28  12 

40  6 

48  12 

68  4 

90  0 

Area  =  3.55  acres.  Tf  this  area  is  added  to  the  area  of  prob- 
lem 26  we  get  for  the  whole  area  42.907  acres,  which  is  the 
area  of  the  farm  shown  in  the  plot  in  Fig.  105. 


CHAPTER  V. 
DIVISION  OF  LAND. 

89.  Division  of  Triangle. — There  are  two  cases  which 
generally  occur  in  practice.  The  first  is  to  draw  a  line  parallel 
to  one  side  of  a  triangle  to  cut  off  a  certain  fraction  of  the 
whole  area,  or  to  divide  the  triangle  into  two  parts  whose  areas 
shall  have  a  certain  ratio,  while  the  second  is  to  draw  a  line 
from  one  of  the  vertices  of  the  triangle  to  divide  it  in  a  given 
ratio. 

First  Case-'  Given  the  triangle  ABC,  Fig.  41,  the  length  of 

whose  sides  is  known.     The  area  of  the  triangle  can  be  found 

from   Formula  3.      It    is    required   to    draw    a   line   PQ   parallel 
to  BC  so  that 


\ 


B  K  C 

Fig.    41.  Fig.    42. 

APQ  :  ABC  ::  m  :  n. 
Let  AP=x,  and  A'Q=y.     Then, 
APQ  :  ABC  : :  AP2  ':  AB2.     .'.    APQ  :  ABC  : :  x2:  c2. 


UT 

In  same  way,      y  =  o+l  — 
Example:     Given  a=  300,  b=  240,  c=  180.     Find  a  line  PQ 

that  will  cut  off  4/9  of  the  triangle  ABC.      *=240-     4/9  = 


240x2/3  =  160.     y  =  180X-2/3  =  120. 

81 


82  SURVEYOR'S    HAND    BOOK. 

Second  Case:  Given  the  triangle  ABC,  Fig.  42,  to  draw  a 
line  AK,  so*  that  AK  will  cut  off  the  triangle  AKB  equal 
to  m/ n  of  the  triangle  ABC.  The  triangles  ABK  and  ABC  have 
the  same  altitude,  and  are  therefore  to  each  other  as  their  bases. 
Hence, 

ABK  :  ABC  : :  m  :  n.     But  ABK  :  ABC  : :  BK  :  BC 
.'.BK  :  BC  : :  m  :n.  BK  =  BC  X  m/n 

EXAMPLE:  Find  BK  in  the  foregoing  example  when  B.  IK  is 
three-fifths  of  the  triangle  ABC.  BK  =  3/o  X  300  =  1  So. 

PROBLEM  51. — Given  a  =  340,  6  =  272,  r  =  '_!'>!.  Find  the  area 
of  ABC  and  AP  and  AQ  when  PQ  is  parallel  to  BC  and  the 
triangle  APQ  is  two-thirds  of  ABC. 

90.  Division  Line  Through  Internal  Point. — It  may  be 
possible  that  it  is  desired  that  the  dividing  line  shall  pass  through 
some  point  inside  the  triangle  and  divide  the  triangle  in  a  cer- 


A  Q    D  B 

Fig.    43. 

tain  ratio.  Let  P  be  the  internal  point  in  the  triangle  ABC, 
Fig.  43,  and  let  it  be  required  to  pass  a  line,  HPQ,  through  P 
that  will  make  the  triangle  AHQ  have  the  ratio  of  m  to  n  to  the 
triangle  ABC.  The  point  P  is  known,  and  the  perpendiculars 
PD  and  PE  are  known,  or  can  be  calculated.  Let  the  area 
of  the  triangle  ABC  be  represented  by  K,  and  PD  =  />,  PE  —  q, 
AQt=x,  and  AH  =  y.  We  have, 

Area  APQ  =  U  PD  X  AQ  =  U  fix 

Area  APH  =  U  PE  X  AH  =  %  qy. 

Area  APQ  +  area  ^P//  =  area  AHQ  = 
*&  (px  +  qyl  =m/n  K   (12) 

Also,  we  have, 

Area  AHQ  =  %  AQ  X  AH  sin.  A  =  %  xy  sin.  A. 

Area  ABC=      AB  X  AC  sin.  A  =  %  be  sin.  A. 


DIVISION    OF    LAND.  83 

But  Area  AHQ  =  m/n  area  ABC 
•-* »  \  xy  si-n-  A  =  2"  be  sin  A 

. ' .  xy  =  m/n  be (13) 

Thus  we  have  two  equations  in  x  and  y,  and  these  can  be 
found  and  laid  off  on  the  sides  AB  and  AC. 

EXAMPLE  :  Given  AB  =  420,  AC  =  400,  BC  =  260,  PD  =  100, 
PE  =  60.  Find  x  (AQ)  and  y  (AH},  when  triangle  AHQ  is 
four- tenths  of  ABC. 

By  calculation  we  find  area  ABC  —  50,400. 
Then  we  have, 

50  x  +  30  3'  =  4/10  50,400  =  20,160. 
A-y  =  4/10  X  420  X  400  =  67,200. 
Solving  for  *  and  y,  we  get, 

.*•  =  219.57  or  183,63; 
y  =  300.05  or  365.75. 


Fig.   44. 

PROBLEM  51. — In  the  triangle,  find  x  and  y  if  the  line  HQ  is 
to  pass  through  P  and  bisect  the  triangle  ABC.  Answer, 
A- =  366.47,  3.  =  229.21. 

91.  Division  of  Quadrilateral. — Given  a  quadrilateral 
ABCD,  Fig.  44.  Required  to  find  a  line  HQ  through  an  inter- 
nal point  P  that  will  make  ADHQ  equal  to  m/n  of  ABCD.  Let 
S  =  area  of  ADHQ  and  K  =  area  ABCD.  The  point  P  is  lo- 
cated by  perpendiculars,  PE  and  PF,  on  two  sides  of  the  quadri- 
lateral. Produce  two  opposite  sides  AB  and  CD  to  intersect  in 
some  point  O.  Let  PF  — />,  PE==q.  The  sides  and  angles 
of  the  quadrilateral  ABCD  are  known,  and  from  these  the  sides 


84  SURVEYOR'S    HAND    BOOK. 

and  area  of  OAD  can  be  calculated.  Adding  area  of  OAD 
to  ADHQ  will  give  the  required  area  of  OHO,  and  adding  the 
area  of  OAD  to  the  area  of  ABCD  will  give  the  area  of  OBC. 
Find  the  ratio  of  OHQ  to  OBC.  The  problem  is  then  re- 
duced to  that  of  finding  a  line  through  P,  dividing  the  triangle 
OBC  into  the  ratio  of  m  to  n.  The  solution  comes  under  the 
case  of  dividing  a  triangle  by  a  line  through  an  internal  point. 
After  the  areas  of  AOD,  OBC  and  OHQ  are  found  we  have, 
where,  OA=a,  OD  =  b.  PE  =  q.PF  =  p,  AQ  =.r.  DH  =  y, 

tt  f>(a  +  x)  +  %  q(b  +  y)  =  area   OHQ, 

(a  +  x)    (b  +  y)  =  m/n  OB  X  OC. 

From  these  two  equations,  the  values  of  x  and  y  can  be  cal- 
culated. In  the  same  way  we  can  find  the  line  passing  through 
an  internal  point  in  a  pentagonal  field,  dividing  the  field  in  a 
certain  ratio. 

PROBLEM  52.— If  .45  =  300,  £C=192,  CD«=144,  AD  =  18Q, 
D£  =  240,  P£  =  96  and  PF  =  60,  find  the  values  of  .v  (  —  AQ) 
and  3?  (  =  DH)  when  the  area  ADHQ  is  seven-twelfths  of 
ABCD. 

92.  General  Solution. — There  are  many  problems  in  land 
dividing  that  can  be  solved  by  special  methods,  and  there  are 
often  short  operations  that  can  be  applied  at  once.  In  the  ma- 
jority of  cases  the  line  of  division  is  not  required  to  pass 
through  an  internal  point.  Where  some  certain  point  is  given 
as  the  point  of  beginning  of  the  division  line,  this  point  is  gen- 
erally at  a  corner  of  the  field  or  on  one  side  at  a  given  dis- 
tance from  a  corner.  In  such  cases  it  is  desired  to  find  the 
bearing  and  length  of  the  dividing  line,  and  this  problem  is 
treated  in  a  general  way  in  the  following  articles.  However, 
no  attempt  is  made  to  solve  problems  of  division  in  regard 
to  the  regular  geometrical  figures,  as  such  solutions  are  raiher 
simple  and  offer  no  difficulties  to  the  student. 

We  have  seen  that  the  sum  of  the  northings  and  the  sum 
of  the  southings  for  a  complete  survey  must  each  equal  zero. 
Thus,  we  have  two  conditions  to  fulfill' and  mathematically  this 
gives  us  two  equations.  If  we  let  A,  h,  /s,  etc.,  represent  the 


DIVISION    OF    LAND. 


lengths,  xind  Bi,  Bz,  B3,  etc.,  represent  the  bearings  of  the  dif- 
ferent courses,  we  must  have  : 

/i  Cos  Bi  +  l*  Cos  £2  +  /3  Cos  B3  etc.  =  0  ......  (14) 

li  Sin  5i  +  I.  Sin  B2  +  /3  Sin  B3  etc  =  0  .....  '.  (15) 

Theoretical!}',  if  we  know  all  the  parts  except  two  we  can 
find  these  two  unknown  parts  from  equations  14  and  15.  The 
lost  or  unknown  parts  can  be: 

Case  I.     Bearing  and  length  of  one  course. 

Case  II.     Length    of   two   courses. 

Case  III.     Length  of  one  course  and  bearing  of  another. 

Case  IV.     Bearing  of  two  courses. 

93.  Case  I.  —  If  the  bearing  and  length  of  one  course  is 
unknown,  the  latitudes  and  departures  of  the  known  courses  are 
first  found.  The  algebraic  sum  of  these  must  be  the  latitudes 
and  departure  of  the  unknown  course  with  the  signs  changed. 

If  we  let  L  and  D  be  the  latitude  and  departure  of  the  un- 
known course,  respectively,  then  the  length  of  the  course 


Dep. 
8.17 
4.49 

(—7.20) 
—5.46 


D 
And  the  tangent  of  the  bearing— y- 

EXAMPLE:     Find  the  lost  parts  in  the  following: 

Course.                      Bearing.                Dist.  Lat. 

AB N62TE                  9.24  4.32 

EC S36°5'E                    7.62  —6.16 

CD (S45°29'W)           (10.10)  (—7.08) 

DA N31°28'W             10.46  8.92 

L=4.32  +  8.92 — 6. 16=7.08 
p=S.  17  +  4.49—5.46=7.20 
Length  CD  =  v/(7.08)-  +  (7.20)2=10.10 

7  *>0 
Tangent  bearing^— ^==1. 1070 

.-.  Bearing=S45°29'W 

PROBLEM  54. — Find  the  lost  parts  in  the  following: 
Course.  Bearing. 

AB N46°22'E 

BC f 

CD S42°W 

DA..  .  N29°W 


Distance. 
38  chains 

42  chains 
54  chains 


80  SURVEYOR'S    HAND    BOOK. 

94.  Case  II. — If. two  lengths  are  unknown  we  first  find 
the  latitudes  and  departures  of  the  known  courses. 

Let  x  and  y  be  the  unknown  lengths  and  M  and  Ar  be  the 
bearings  of  these  courses,  respectively.  Then  from  equations 
14  and  15  we  have: 

%  Cos  M+y  Cos  N  +  L=0 
x  Sin  M  +3  SinN  +  D=0 

Multiply  the  first  equation  by  Sin  N  and  the  second  by  Cos  TV 
and  we  have : 

x  Cos  M  Sin  N  +  y  Cos  N  Sin  N  +  L  Sin  N=0 
x  Sin  M  Cos  X+y  Cos  N  Sin  N  +  DCos  N=*0 

Subtracting  and  transposing,  we  get : 

x  (Sin  M  Cos  N—Cos  M  Sin  N)=L  Sin  N—D  Cos  N 
x  Sin  (M—N)=L  Sin  N—D  Cos  N 
LSinN—D  CosN 

Sin  (M-N) 

Example. — Find  the  lost  parts  in  the  following  survey : 
Course.         Bearing.          Dist.  Lat.  Dept. 

AB  ...         N47°2'E         31.30  21.33  22.90 

BC  ...        S57°4'E         21.10  —11.47  1771 

CD  ...         S60°W  x  —x  Cos  60°          -x  Sin  60° 

DA  ...        N40°W  y  y  Cos  40°          — y  Sin  40° 

From  formulas  (14)  and  (15),  we  get, 

'  —  xCosQ()°  +  y  Cos  40+   9.86=0 
—  xSm  60  — >>  5m  40 +  40. 6 1=0 

Multiplying  the  first  equation  by  Sin  40°  and  the  second  by 
Cos  40°  we  have : 

—x  Cos  60°  Sin  40° +  y  5*n  40°  Cos  40°+    9.86  5m  40°=0 
— x  Sin  60°  Cos  40°— y  Sin  40°  Cos  40° +  40.61  Cos  40°=0 
Transposing  and  changing  signs  we  have : 
x  Cos  60   5m  40°— y  Sin  40°  Cos  40°=  9.86  5-m  40 
x  Sin  60°  Cos  40°  +  y  Sin  40°  Cos  40°=40.61  Cos  40 
Adding : 

x  (Sin  60°  Cos  ±0°  +  Cos  60°  Sin  40°)=40.61  Cos  40°  +  9.86  Sin  40 
x  Sin  100°=40.61  Cos  40° +  9.86  5m  40° 
40.61  Cos  40°  +  9.86  Sin  40° 


5m  100° 
4061  X  .76604  +  9.86  X  .64279 
.98481 


=38.024 


DIVISION    OF    LAND. 


87 


If  we  multiply  the  first  equation  by  Sin  60°  and  the  second 
by  Cos  60°  we  get: 

x  Cos  60°  Sin  60°— y  Sin  60°  Cos  40°=  9.86  Sin  60° 

x  Cos  60°  Sin  60°  +  y  Cos  60°  Sin  40°=40.61  Cos  60° 

Subtracting  and  changing  the  signs  we  have : 

y  (Sin 60°  Cos  40°  +  Cos  60°  Sin  40°)=40.61  Cos  60°— 9.86  Sin  60° 

y  Sin  100°=40.61  Cos  60°— 9.86  Sin  60° 

'  ^40.61  Cos  60°— 9.86  Sin  60° 

Sin  100° 
40.61  X.5--9. 86  X. 86603 


PROBLEM 
irse. 

y 
55.  —  Find 

.98481 
the  lost  parts. 

Bearing. 
N5°E 

.   S17°E 

.  .  .  .   S56°E 

(.. 

.  N85°W 

9.58 


Course.  Bearing.  Distance. 

\B N5°E  8.68 

/>>r S17°E  x 

(7). 
DA 

.r  =  4.087,  .v  =  8.937 

95.     Case  III. — The   length   of  one   course  and  the   bear- 
ing of  another  lost. 

Find  the  unknown  parts  in  the  following  example: 


Course. 
AB  
BC  
CD  

DE  ...  . 
EA  .  .  . 

Bearing. 
X36°E 
X° 
S20°E    . 
S75°W  v 
N30°W 

Distance 
12  chains 
8  chains 
11  chains 
y  chains 
10  chains 

Latitude. 

9.708 
—  8  Cos  X° 
—10.337 
—  y  Cos  75° 
8.660 

Departure. 
7.054 
8  Sin  X° 
3.762 
—  y  sin  75° 
—5.000 

In  all  cases  it  is  better  to  make 
a  graphical  solution  in  order  to 
find  the  direction  letters  of  the 
bearing.  Lay  off  AB,  Fig.  45, 
N.  36°  E.,  equal  to  12  chains  to 
some  scale,  anc!  EA  S.  30°  E. 
10  chains.  C  will  be  somewhere 
on  the  circumference  of  a  circle 
whose  center  is  B  and  whose 
radius  is  8  chains,  while  D 
will  be  somewhere  on  ED, 
N'T  ere  ED  is  ^.rawn  with 


Pig. 


88  SURVEYOR'S    HAND    BOOK. 

a  bearing  of  N.  75°  E.  Through  B  draw  BD'  S.  20°  E.,  and 
lay  off  CD"  from  D'  equal  and  parallel  to  CD.  Through  C' 
draw  CC  parallel  to  ED  and  cutting  the  circle  at  C  and  C"  and 
through  C  and  C"  draw  CD  and  CD"  parallel  to  BD'.  There 
are  two  solutions,  ABCDE  being  one  and  ADC"D"E  being  the 
other.  From  the  figure  we  see  that  the  bearing  of  BC  is  south- 
east, and  that  of  BC"  is  southwest.  Filling  out  the  table  for 
the  southeast  bearing  and  adding  the  latitude  and  departures, 
we  get: 

8  Cos  X°  +  y  Cos  75°  =  +  8.031 
8  Sin  X°—y  Sin  75°=  —  5.816 

Multiplying   the    first    equation   by   Sin  75°    and   the    second 
by  Cos  75°,  we  have: 

8  Cos  X°  Sin  7.5°  +  y  Cos  75°  Sin  75°  =  8.o31  Sin  75° 
8  Sin  X°  Cos  75°  —  y  Cos  75°  Sin  75°  =  —  5.816  Cos  75° 
Adding,  we  have: 

8  (Sin  X°  Cos  75°  +  Cos  X°  Sin  75°)  = 
8.031  Sin  75°  —  5.816  Cos  75° 


Sin  <X°  +  75°)  =    -.  _  .78161 

O 

X°  +  75°  =  128°36'  or  51°24' 
A'0=53°36'  or  23°36' 
To  eliminate  X°,  we  have  : 

8.031  -  y  Cos  75° 

U  OS  A.     ~  o 

'  ' 


Squaring  and  adding,  we  have  : 

64  =  98.322817  —y  (16.062  Cos  75°  +  11.632  Sin  75°)  +  / 
Ay8  —  y  (11.632  Sin  75°  +  16.062  Cos  75°)  =  —  34.322817 
y2  —  15.342864  y  =  —  34.322S  17. 
Completing    the    square,    we    have: 
y=  1-2.685,  or  2.705. 

PROBLEM  55.  —  Find  the  bearing  X  and  the  distance  37  in  the 
preceding  examples,  when  the  course  BC  bears  southwest. 


DIVISION    OF    LAND.  89 

Answer.     Rearing,   23°36.     Length.   2.704. 
96.     Case    IV.  —  T\v<>    bearings    unknown. 
Let  X  and   Y  be  the  unknown  bearings,  a  and  b  the  lengths 
of  these   courses,  and  L  and   D  be   the   latitude   difference  and 
the    departure    difference    of    these    courses,    respectively;    then 
a  Cos  X°  +  b  Cos  Y°  =  L 
a  SinX°  +  b  Sin  Y°=D 

D-aS*nX 


Then  Cos  Y*  -  ^  ^  yo 

Squaring  and   adding,  we  have  : 


Let  27'  =  a2  +  L2  +  D2  —  bz 
"But  Cos2  X=  \-Siri2  X 


Therefore  1  -  5»,'  X  = 


Then  a2L2—  P  =  a2  (L2  +  D2)  Sin2X  —  2aDT  Sin  X 
From  this  quadratic  in  Sin  X  two  values  of  Sin  X  will  be 

found  and  there  will  be  two  solutions  possible. 

97.     Example.  —  Find    the    unknown    parts    in    the    follow- 

ing example  : 

Course.  Bearing.  Distance.  Latitude.  Departure. 

AB.....  N24°E  26  chains  23.752  10.575 

BC  .....  Sx°E  28  chains  —28  Cos  X  28  Sin  X 

CD  .....  S38"E  24  chains  —18.912  14.776 

DE  .....      Sy°W  36  chains  —36  Cos  Y  —36  Sin  Y 

EA  .....  N44°W  18  chains  12.948  —12.504 

To  find  the  direction  letters  draw  AB,  Fig.  46,  N.  24°  E., 
and  EA  S.  44°  E.,  move  CD  from  its  true  position  to  some  po- 
sition C'D'  parallel  and  equal  to  itself  where  C'  coincides  with  B. 
C  has  been  moved  28  chains,  because  the  length  of  BC  is  28 
chains.  Now,  D  is  28  chains  from  D',  but  D  is  also  36  chains 
from  E,  hence  with  D'  as  a  center  and  a  radius  of  28  chains 
describe  an  arc,  and  with  E  as  a  center  and  36  chains  as  a 
radius  describe  an  arc  cutting  the  first  arc  at  D.  Draw  DC  N. 
38°  W.  24  chains.  Draw  BC  and  DE.  Thus,  we  see  that  BC 
bears  southeast  and  that  DE  bears  southwest.  Putting  the  di- 


90 


SURVEYOR'S    HAND    BOOK. 


rection  letters   in  the  table  and  filling  out  the  latitude  and  de- 
parture columns,  we  have  for  our  equations: 
2S  Cos  X  +  30  Cos  F=  17.788 
X  —  30  Sin  7  =  —  12.847 


4.447  —  7  Cos  X  3.2  12  +  7  Sin  X 

Then  Cos  Y  =  -  g  -  and  S^n  Y=  --  —  - 

81  =  30.092753  —  62.258  Cos  X  +  44.968  Sin  X  +  49 
4.447  Cos  X  =  3.212  5m  X  —  .130232 
Co*  X  =  .  72228  Sm  X  —  .030035 

1  —  Sm'  X  =  .009285  +  .04425409  Sin  X  +  .521G884  SVn8  X 
Sin2  X  +  .02908  Sin  X  =  .65655 
SIM  X=  82498 
X  =  50°35'll" 

PROBLEM   50.—  Fnd   the  lost  parts. 

Course.  Rearing.  Distance. 

AB  ................................   X31°E  14  chains 

BC  ................................   N62°E  20  chains 

CD  ................................        x  •_'"  chains 

DE  ................................   S38°W  23  chains 

EA  ................................       y  24  chains 

98.  Dividing  Land.  —  It  oft- 
en becomes  necessary  to  di- 
vide farms  among  the  differ- 
ent owners.  A  certain  number 
of  acres  is  sold  from  one  part 
of  a  farm,  and  it  becomes  nec- 
essary to  know  the  boundaries 
of  the  part  cut  off  from  the 
original  survey.  The  partition 
is  generally  made  in  two 
ways,  either  by 


IG. 


a  line  starting  at  a  certain  point  cutting  off  the  required  number 
of  acres,  or  by  a  line  that  has  a  certain  bearing.  The  following 
examples  will  serve  to  illustrate  the  methods. 

99.  Example. — Find  the  bearing  and  length  of  a  line 
AP  that  will  cut  off  40  acres  from  the  farm  ABCD,  as  given  be- 
low in  Fig.  47. 


DIVISION    OF    LAND. 


01 


Course. 
AB  
BC 

Bearing. 
N47°2'E 
S57°4'E 
S28°42'W 
N40°27'W 

Corrected 
Departure. 

22.71 
17.6-5 
—  19.17 
—21.19 

Distance. 
31.0  chains 
21.0  chains 
40.0  chains 
32.7  chains 

D.  M.  D. 

22.71 
63.07 
61.5* 

21.19 

Lat. 
21.13 

—  11,12 
£-'35.09 

24.88 

Corrections. 
Dep.       Lat.    Dep. 
22.68       .13       .03 
17.63       .08       .02 
—  19.21        .16       .04 
—  21.22       .13       .03 

CD  
DA  

Corrected 
Latitude. 

21.26 
—  11.34 
—  34.93 
25.01 

—  46.51 
+  46.01 

—  40.43 
40.31 

—      .50 

/ 

482.8146 

—    .12 
Lrea. 

•  715.2138 
2149.9415 

529.9619 

1012.7765 


2865.1553 
1012.7765 


Double  area  =  1852.3788  sq.  ch. 
Area=     92.61894  acres. 


Join  the  starting  point  A  of  division  with  the  corner  C  near- 
est the  final  end  of  the  required  course.     Find  the  area  of  the 
part   thus   cut   off  as    follows : 
Course.  Cor.  Lat.     Cor.  Dep.     D.  M.  D.         Area. 

AB -21.26  2-2.71  22.71  482.8146 

BC — 11.34  17.65  63.07        —  715.2138 

CA -9.92       —40,36  40.36         -400.4712 

Area  =  31. 64352  acres. 

As  the  area  of  the  triangle  ABC  is  only  31.64  acres,  the  line 
AP  that  makes  area  ABCP  equal  to  40  acres  must  cut  the  side 
CD.  hence  P  lies  on  side  CD. 

Length  CA  =  x/(9.92)2  + (40.36)2=  41.561 
40.36 

Tan.  bearing  of  CA  =  "9-92"  ~  4-06855 

.'.  Bearing  of  CA  =  76°  11 '28" 

Angle  ACP  =  47°29'28" 

Now  area  ACP  =  400  —  316.4352  =  83.5648  sq.  chains. 

But  area  ACP  =  ^  CA,  CP  sin.  ACP 

2  area  ACP  1671296 

* '  •  CP  ~  ^  417561  x  .73717  ""  5'465  chaills 


92  SURVEYOR'S    HAND    BOOK.    . 

The  latitude  and  departure  of  CP  bear  the  same  ratio  to  the 
corrected  latitude  and  departure  of  CD  that  the  length  Cl3  does 
to  CD. 

.*.  Lot.  CP  =  4.785  r>ct>.  CP  =  2.0-2' 

To  find  the  length  and  bearing  of  PA,  complete  the  table 
of  ABCP. 

Course.                      Latitude.  Departure.     D.  M.  D.          Area. 

AB 21.26  22.71          22.71               482.8140 

BC — 11.34  17.65           03.07  _  715.2138 

CP -4.785  -    2.02           78.10  —374.4085 

PA —  5.135  —  37.74          37.74  — 103.7949 

Double  area  =  800.6026  square  chains. 
Area  ABCP  —  40.03  acres. 

PROBLEM  57. — In  the  'example  in  Article  99,  find  the  bearing 
and  length  of  a  line  AP  that  will  cut  off  an  area  ABP  equal  to 
nine  acres. 

PROBLEM  58. — Find  the  bearing  and  length  of  a  line  DK  in 
the  preceding  problem  that  will  make  area  ADK  equal  to  six 
acres. 

100.  Example. — Find  the  length  of  a  line  that  bears  N. 
52°  F.  and  cuts  off  51  acres  on  the  northwest  side  of  the  farm 
ABCD  above. 

Draw  a  line  CP,  Fig.  47,  through  C  that  bears  N.  52°  E., 
and  find  the  length  CP  and  AP. 

Applying  equations   (14)  and  (15)  we  get: 

x  cos.  40°27'  —  y  cos.  52  =  — 9.92  (A) 

.r  sin.  40°27'  +  y  sin.  52  =40,36  (B) 

Eliminating  y 

40.36  cos.  52—9.92  sin.    52 

~sin.  92°27' 
Similarly, 

40.36  cos.  40°27'  -f-  9.92  sin.  40°27'     ^  ,  fto 

-    yr—  sin.  92°27' =S7'193 

Find  the  area  of  A  B  C  P,  as  follows: 


DIVISION    OF    LAND. 


Course.  Latitude. 

AB 21.26 

BC —11.34 

CP —22.89 

PA..  .  +12.97 


Departure. 
22.71 

17.65 
—  29.30 
— 1  1.06 


D.  M.  D.  Area. 

22  71  482.S146 

63.07  -  715.2138 

51.42  - 1177.0038 

11.06  143.4482 


Area  ABCP  =  63.29724  acres. 


Fig.    47. 

The  line  CP  cuts  off  12.29724  acres  in  excess.  Let  the  line 
MN,  parallel  to  CP.  cut  off  the  required  area.  Hence  the  area 
MNCP  is  122.9724  square  chains.  From  C  and  P  drop  perpen- 
diculars on  MN,  cutting  it  at  K  and  H. 

Angle  MPH=2'Z?';  angle  A'C/fT=1904' 
Let  r:  =  altitude  of  trapezoid  MNCP  =  PH  =  CK 
Now, 

MNCP  =  HKCP  —  NCK  +  MPH 


94  SURVEYOR'S    HAND    BOOK. 

.'.    122.9724=37.193  z  -  ^  tan.  19°4'+  —tan.  2°27' 
2.  2i 

^-(tan.  19°4'— /aw.  2°27')  — 37.193  s=— 422.9724 

.15305s2— 37.1930-=— 122.972* 
z2— 243. 01 2s=— 803.48 

.'.£  =  3.353  chains 

NC  =  3.353  +  cos.  19°4' =3.548 

PM  —  3.353  -r-  c  <w.  2°27'  =  3.356 

iV  A' =  3.353  tan.  19°4'=1.16. 

MH  =  3.353  tan.  2°27'  =  .14 

The  field  notes  of  the  51  acres  will  read  as  follows : 

Course.                                                       Bearing.  Distance. 

AB N47°2'E  31.0      chains 

BN S57°4'E  17.452  chains 

NM S52°W  M.173  chains 

MA N40°27'W  13.690  chains 

PROBLEM  57. — Find  the  bearing  and  length  of  a  line  that 
starts  from  mid-point  of  CD  and  bisects  farm,  Fig.  47. 

PROBLEM  58. — Find  bearing  and  length  of  a  line  that  starts 
.on  AD  15  chains  from  A  and  cuts  off  50  acres  from  west  side 
of  farm,  Fig.  47. 

PROBLEM  59. — In  the  example  of  Fig.  '47  find  the  bearing 
and  length  of  a  line  that  starts  at  a  point  H  on  AB  15  chains 
from  D  and  bisects  farm. 

PROBLEM  CO. — Find  bearing  and  length  of  a  line  DP  in  ex- 
ample of  Fig.  47  that  cuts  off  3  acres  on  left  of  dividing  line. 

PROBLEM  61.— Find  the  length  of  line  PQ  that  bears  N  45°  W. 
and  bisects  farm  in  example  of  Art.  100. 

BIBLIOGRAPHY. — The  works  of  the  late  ].  B.  Johnson  and  the 
late  Charles  Davies,  which  have  already  been  described,  have 
sections  that  deal  with  the  problems  of  land  dividing. 

"Plane  Surveying,"  by  Daniel  Carhart,  gives  not  only  a  treat- 
ment of  the  land  division,  but  also  of  the  theory  and  use  of 
instruments  and  methods  of  surveying,  calculation,  earthwork, 
etc.,  tables  "A  Treatise  on  Surveying,"  Part  I,  by  the  late  W. 


DIVISION    OF    LAND.  [)5 

M.  Gillespie,  restricts  its  discussion  to  land  surveying  and  di- 
rect leveling,  and  under  the  subject  of  land  division  it  gives 
a  great  number  of  problems  for  the  division  of  land,  illustrated 
by  figures  and  examples. 


CHAPTER  VI. 
LEVELING. 

101.  The   Y    Level — The   essential    parts   of   a   Y    level, 
Fig.  48,  are  the  bubble  tube  and  the  line  of  sight     The  latter 
is  determined  by   the  telescope  and   should   be  parallel  to   the 
axis  of  the  bubble  tube.    The  telescope  rests  in  two   T-shaped 
supports   called    }'rs  or   Wyes,   which   are  attached   to   a   hori- 
zontal bar.    The  lower  part  of  the  wye  is  formed  into  a  threaded 
bolt  that  passes  through  a   hole  in  the  end  of  the  horizontal 
bar.     Two  capstan   nuts  are   attached  to  each   wye.  one  above 
and  one  below  the  bar.     By  turning  these  capstan   screws   the 
wye  can  be  raised  or  lowered  at  pleasure.      Small,  hard,  steel 
pins,   about    1-HJ   in.    in   diameter,  are   used    for   operating   the 
capstan    screws.    The   horizontal   bar   is   attached   by   a   screw- 
joint  to  a  vertical  axis  turned  in  the  form  of  a  frustrum  of 
a  cone.     The  telescope  with  the  wyes,  horizontal  bar,  and  socket 
can  be   removed  from  the  level-head.    The   level-head  consists 
of  a  horizontal  brass  plate  enlarged  into  a  ball  and  socket  joint 
in  the  center  and  into  a  rim  with  screw  threads  on  the  circum- 
ference; the  former  is  to  provide  an  adjusting  motion  for  the 
leveling   screws,  and   the   latter  for   attachments   to  the   tripod 
head.    Above  the  brass  plate,  which  is  attached  to  the  tripod,  is 
another   plate   provided    with    four   vertical,  .cylindrical   screws, 
into   which   the   four   leveling  screws   rest   in   small    seats    with 
ball  and  socket  joints,  and  are  operated  by  milled-head  screws. 
A  longitudinal  cross-section  of  the  level  and  telescope  is  shown 
in  Fig.  49. 

102.  The    Telescope. — The    telescope,    Fig.    49,    consists 
of  an  eye-piece,  an  objective,  and  a  tube  to  hold  them  in  place. 
The  eye-piece  is  a  very  small  microscope  of  a  very  short  length, 
one  end  of  which  is  very  near  the  cross  wires.     In  the  erecting 
telescope  it  consists  of  four  lenses:  the  eye  lens,  the  field  lens. 
the  amplifying  lens,  and  the  image  lens,  arranged  in  "order  from 
the  eye.    The  objective  consists  of  a  special  tube  sliding  in  the 

96 


LEVELING. 


98  SURVEYOR'S    HAXD    BOOK. 

main  barrel  of  the  telescope  with  a  double  lens  in  the  outer 
end.  The  objective  is  held  true  to  its  place  by  two  collars  in- 
side the  main  tube.  The  lens  has  a  long  focal  length  and 
draws  the  image  to  the  plane  of  the  cross  wires.  If  this  lens 
were  a  double  convex  lens  it  would  neither  bring  the  rays  to 
an  exact  focus  nor  make  them  colorless.  Hold  a  double  con- 
vex lens  so  that  its  central  plane  is  perpendicular  to  the  rays 
of  the  sun  and  hold  a  sheet  of  paper  back  of  the  lens  and  move 
it  to  and  fro  to  find  the  focus.  If  the  paper  is  held  between 
the  focus  and  the  lens  the  edge  of  the  bright  circle  will  be 
colored  red.  Move  the  *  paper  beyond  the  focus  and  we  find 
the  edge  colored  blue.  In  any  lens  all  parallel  rays  of  sun 
light,  having  equal  wave  lengths,  are  brought  to  a  focus  at  a 
fixed  distance  behind  the  lens,  called  the  focal  length,  or  the 
principal  focal  distance. 

If  the  lens  in  the  end  of  the  objective  were  single,  the  rays 
of  sunlight  would  not  be  brought  to  a  focus,  but  the  object 
would  be  fringed  with  colors ;  that  is,  the  single  lens  makes 
the  rays  planatic  (wandering)  and  chromatic  (colored).  To 
make  the  rays  aplanatic  and  achromatic  the  object  glass  is  m;i<Ki 
of  two  lenses,  Fig.  50,  a  double  convex  and  a  plano-concave ; 
the  former  of  crown  glass  and  the  latter  of  flint  glass.  The 
refractive  indices  of  these  kinds  of  glass  supplement  each  other 
and  the  rays  are  brought  to  a  focus  and  are  colorless. 

The  eye-piece  is  moved  by  means  of  milled-hcad  screws  at- 
tached to  a  rack  arnd  pinion  movement,  or  by  a  spiral  slot  into 
which  a  pin  works.  In  the  first  case  the  eye-piece  is  moved 
by  the  milled-head  screws  until  the  cross  wires  come  into  view  ; 
in  the  latter  case  the  eye-piece  itself  is  moved  backward  and 
forward  in  the  telescope  by  turning  it.  The  cross  wires  are  at- 
tached to  a  brass  ring,  called  the  reticule,  which  is  controlled  by 
small  capstan  screws  outside  the  telescope. 

The  tripod  is  a  three-legged  support  connected  to  a  plate  to 
which  the  level-head  is  screwed  when  the  instrument  is  in  use. 
The  legs  are  made  of  hard,  straight-grained  wood,  and  shod 
with  hard,  steel  conical  shoes. 


LEVELING. 


99 


100  SURVEYOR'S    HAND    BOOK. 

103.  Setting    Up    the    Instrument. — Set    the    tripod    with 
legs  well  spraddled,  and  then  place  the  level  on  the  tripod,  screw- 
ing   the    level-head    into    the    tripod    cap.     Bring    the    telescope 
parallel  to  two  opposite  leveling   screws ;   turn   the  screws  both 
out   or   both   in.   making   the   left   thumb  move   in   the   direction 
that  the  bubble  is  to  shift.     After  the  bubble  reaches  the  center 
of   its  turn,  turn  the   telescope  over  the  other  pair   of  opposite 
screws    and    repeat    the    left-thumb    process.     Repeat    and    check 
both  on  second   leveling. 

104.  Rods. — Leveling   rods   used   by    engineers    are   divid- 
ed into  feet,  tenths  of  a  foot,  and  hundrcdths  of  a  foot,  and  have 
a  vernier  attachment,   which   enables  the  rod  to  be  read  to  the 
thousandth  part  of  a  foot. 

The  Philadelphia  rod,  Fig.  51,  is  usually  IVz  ft.  long,  and 
is  made  in  two  pieces,  which  may  be  effectively  extended  to 
a  length  of  12  ft.  This  rod  has  the  foot  division  lines  marked 
by  red  figures;  the  even  tenths  of  a  foot  are  marked  by  black 
figures ;  and  each  alternate  hundredth  of  a  foot"  is  painted  black 
half  way  across  the  rod  on  a  white  background.  This  enables 
the  rod  to  be  read  to  the  nearest  hundredth  of  a  foot  from  a 
distance  through  the  telescope  by  the  levelman.  When  the  rod 
is  extended,  a  continuous  graduation  to  12  ft.  is  visible.  This 
rod  is  provided  with  a  target,  a  circular  piece  of  metal  about 
4  ins.  in  diameter,  alternate  graduations  of  which  are  painted 
red  and  white.  The  target  slides  along  the  rod,  and  its  exact 
distance  from  the  end  of  the  rod  may  be  read  by  means  of  a 
hole  in  the  center.  A  vernier  attached  to  the  target  enables 
the  rodman  to  read  to  the  thousandth  part  of  a  foot.  This 
rod  is  intended  for  quick  work  and  hard  service.  It  should  be 
made  of  the  best  wood,  brass  trimmings  and  varnished  to  re- 
sist water. 

The  New  York  rod,  Fig.  52,  is  similar  to  the  Philadelphia 
rod,  but  it  is  lighter  and  much  more  care  is  taken  in  its  gradua- 
tion. The  rod  can  not  be  read  directly  from  the  instrument. 
It  is  intended  for  precise  leveling. 


LEVELING. 


101 


102  SURFEYOR'S    HAND    BOOK. 

Figure  53  shows  a  form  of  self-reading  rod  that  can  be 
used  when  it  is  desired  to  read  the  rod  directly  from  the  in- 
strument. Its  graduations  are  similar  to  those  of  the  Philadel- 
phia .<  cd,  but  it  is  somewhat  lighter.  The  Philadelphia  rod  can 
he  used  as  a  self-reading  rod  and  it  is  often  convenient  to  use 
it  as  such. 

105.  Theory  of  Leveling. — When  an  engineer's  level  has 
been  set  up,  and  the  bubble  brought  to  the  center  of  the  bubble- 
tube,  the  line  of  sights  is  horizontal.  The  elevation  of  this 
horizontal  line  can  be  found  by  noticing  how  much  it  strikes 
above  some  point  whose  elevation  is  known,  and  adding  this 
distance  to  the  known  elevation  of  the  reference  point  or  datum. 
Having  determined  the  elevation  of  the  horizontal  line,  the 


D4-—  __ 


Fig.    54. 

elevation  of  any  other  point  may  be  easily  found  by  noticing 
now  much  the  horizontal  line  is  above  the  point  in  question, 
and  subtracting  this  amount  from  the  elevation  of  the  line  of 
sights.  The  term  "height  of  instrument"  is  given  to  the  eleva- 
tion of  the  horizontal  line  of  sights. 

Suppose  the  elevation  of  some  point  A,  Fig.  54,  has  been 
determined  and  is  100  ft.  above  some  plane  known  as  the  datum 
plane,  the  elevation  of  which  is  called  zero.  It  is  desired  to 
find  the  elevation  of  some  point  P.  Set  the  instrument  at  B 
and  get  the  rod  reading  AD,  which  is  8.46  ft.  Adding  8.46  ft. 
to  100  ft.  gives  108.46  ft.  as  the  height  of  the  horizontal  line 
of  sights,  so  we  say  that  the  height  of  the  instrument  (H.  I.) 
is  108.46  ft.  Sight  next  on  point  C  and  read  the  distance  CK 


LEVELING.  10$ 

on  the  rod,  which. is  2.05  ft.;  subtracting  this  from  the  H.  I., 
108.46  ft.,  gives  us  10(5.41  ft.,  the  elevation  of  the  point  C.  The 
elevation  of  the  point  C  having  been  found,  the  point  C  may 
be  used  to  find  the  elevation  of  another  point  in  the  same  way 
that  the  elevation  of  the  point  A  was  used  to  find  C.  Thus,  set 
the  instrument  at  E  and  read  the  rod  on  C  and  let  CM  =  6.58 
ft.  Then  the  new  H.  I.  =  106.41  +  6.58  =  112.99. 

It  often  happens  that  the  line  of  sights  strikes  the  ground 
in  front  of  a  regular  station  as  at  G.  If  this  occurs,  hold  the 
rod  on  some  intermediate  point  as  F,  and  take  a  rod  reading. 
Tt  is  necessary  in  such  cases  to  select  a  point  that  is  firm  and 
hard.  A  smooth  stone,  firmly  imbedded  in  the  soil,  makes  an 
excellent  point  for  such  purposes.  Suppose  the  rod  reading  on 
such  a  turning  point  was  ArF  =  1.29.  The  elevation  of  F-= 
112.99  —  1.29  =  111.70.  Then  set  the  instrument  at  some  point  H, 
level  up,  and  take  the  rod  reading  again  on  F  (back  sight),  where 
LF—  11.42.  The  height  of  instrument  (H.  I.)  =111.70 -f  11.42 
=  128.12.  The  rod  reading  GR  on  the  <  regular  station  G  =  6.48 
and  the  elevation  is  llll.lU,  while  the  rod  reading  PQ  on  point 
P  h  ;5.82  and  the  elevation  of  P  is  119.80. 

It  will  be  well  to  bear  in -mind  that  a  back  sight  is  a  rod 
reading  taken  on  a  point  whose  elevation  is  known,  and  that 
a  fore  sight  is  a  rod  reading  taken  on  a  point  whose  eleva- 
tion is  unknown.  Always  add  the  back  sights  to  the  elevation 
of  the  point  to  get  the  height  of  instrument ;  and  subtract  fore 
sights  from  the  height  of  instrument  to  get  the  elevation  of 
the  point  on  which  the  fore  sight  was  taken.  The  H.  I.  is 
always  in  the  line  above  the  fore  sight,  and  the  H.  I.  will  not 
be  changed  till  the  instrument  is  moved  to  a  new  position. 

The  starting  point  A,  the  elevation  of  which  has  been  pre- 
viously determined,  is  called  the  Bench  Mark,  abbreviated  B.  M. 
Intermediate  points,  such  as  C  and  F,  are  called  Turning  Points, 
T.  P.  Whenever  possible  rounded  stones,  solidly  imbedded  in 
the  earth  and  almost  covered,  are  the  best  T.  P.'s. 

The  following  is  a  convenient  arrangement  of  column  head- 
ings for  level  notes : 


104  SURVEYOR'S    HAND    BOOK. 

Remarks. 


Station. 
0 
1 
+80 
2 
3 

B.  S. 
8.46 
6.58 
11.42 

H.I. 
108.46 
112.99 
123.12 

F.S. 

2:05 
1.20 
6.48 
3.32 

El. 
100.00 
106.41 
111.70 
116.64 
119.80 

Figure  55  illustrates  a  typical  level  notebook. 
PROBLEM   02.— Fill  out  the  column   for  H.  T.  and  El.  in  the 
table  below : 

Sta.  B.  S.  H.  I.  F.  S.  El. 

26  3.26  Tii.l-J 

27  ...  ....  7.12 

28  1.08  ....  ll.s  I 
20                 ...                ....  5.21 

30  ...  ....  o.«;s 

+72  1.24  ....  11.04 

31  ....  4  .!•; 
32 

33  ...    ,  .....  .  11, VJ 

PROBLEM  63.— If  there  is  a  B.  S.  on  station  27  of  3.22,  fill 
out  a  table  for  the  remaining  H.  I.'s  and  El.'s.  .- 

106.  Bench  Marks. — The  relative  elevation  of  any  num- 
ber of  points  near  each  other  or  widely  separated  may  be  found 
by  comparing  their  heights  above  the  datum  plane.  The  datum 
most  extensively  used  is  mean  sea  level  and  its  elevation  is 
said  to  be  zero.  A  bench  mark  is  a  point,  the  elevation  of 
which  has  been  carefully  and  accurately  measured,  marked  and 
checked,  and  which  may  be  used  as  a  starting  point  for  any 
leveling  that  may  be  contemplated  in  its  immediate  vicinity. 
The  best  form  of  bench  mark  is  a  copper  bolt  firmly  imbedded 
in  masonry,  which  is  not  likely  to  settle.  The  United  States 
Geodetic  Survey  has  established  a  great  many  such  bench  marks 
throughout  the  country.  The  elevation  of  each  of  these  should 
be'  carefully  marked  either  on  the  head  of  the  bolt  or  on  a 
copper  plate  attached  to  the  masonry.  In  running  a  line  of 
levels  across  country  excellent  and  lasting  bench  marks  can  be 
made  by  chopping  away  a  portion  of  a  large  root  of  a  large 
tree  until  the  part  remaining  is  in  the  form  of  a  low,  broad- 
based  pyramid,  and  then  driving  a  nail  or  spike  into  the  vertex. 


LEVELING. 


105 


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106  SURVEYOR'S    HAXD    BOOK. 

107.  Profiles. — A  profile  is  a  drawing  that  shows  the  rise 
and  fall  of  the  ground  on  which  the  line  was  surveyed.  The 
surveyed  line  may  be  straight,  curved,  or  broken. 

To  make  a  profile  elevations  of  points  on  the  line  at  short 
regular  intervals  must  be  found,  as  well  as  the  points  where 
there  is  a  sudden  change  in  the  surface. 

Profiles  are  usually  drawn  to  a  horizontal  scale  of  1"  =  400', 
and  a  vertical  scale  of  1"  =  20'.  Paper  properly  divided  into 
squares  by  horizontal  and  vertical  lines  can  be  purchased  by  the 
roll  or  sheet. 

108. — Crosswire  Adjustment. — To  make  the  intersection 
of  the  cross  wires  intersect  in  the  axis  of  the  telescope  or  the 
line  of  collimation,  set  up  the  instrument,  level,  and  bring  the 
cross  wires  into  view  by  turning  the  telescope  to  clear  sky. 
Focus  the  objective  on  some  wall,  and  then  have  an  assistant 
mark  a  spot  on  the  wall  at  the  intersection  of  the  cross  wires 
with  a  soft  pencil;  loosen  the  clips  or  loops  that  control  the 
telescope,  note  that  it  still  points  to  the  spot  on  the  wall,  then 
turn  the  telescope  in  the  wyes  with  the  right  'hand  until  the 
bubble  tube  is  on  top.  If  the  cross  wires  still  intersect  on  the 
spot  the  instrument  is  in  adjustment;  if  it  intersects  above  or 
below,  loosen  the  small  capstan  screws  that  control  the  wire 
ring  and  turn  them  so  that  the  cross  wire  will  be  moved  back 
one-half  of  the  displacement.  Bring  it  back  to  the  spot  by  the 
leveling  screws,  and  check  by  repeating  the  process. 

To  correct  t"he  vertical  wires  turn  the  telescope  so  that  the 
bubble  is  to  the  right  or  left  of  the  instrument  and  in  the  same 
horizontal  plane,  and  bring  the  cross  wires  on  the  spot  by  the 
leveling  screws,  then  turn  the  telescope  on  its  horizontal  axis 
180°,  and  if  there  is  any  displacement  correct  one-half  by  the 
capstan  screws  that  control  the  vertical  wire  and  the  other  half 
by  the  leveling  screws.  Check  by  repeating  the  process. 

109.  Bubble-Tube  Adjustment. — To  make  the  axis  of 
the  bubble  tube  parallel  to  the  line  of  collimation,  loosen  the 
clips  and  level  accurately,  then  take  the  telescope  in  the  hand 
and  turn  it  end  for  end  in  the  wyes.  If  the  bubble  remains 


LEVELING.  107 

in  the  center  of  the  tube  it  is  in  adjustment,  but  if  it  does  not, 
raise  or  lower  one  end  of  the  bubble  tube  by  means  of  the  small 
capstan  screws  to  correct  one-half  of  the  displacement.  The 
rest  is  corrected  by  the  leveling  screws.  Repeat  until  it  checks. 

Level  accurately,  revolve  the  telescope  slowly  in  the  wyes 
and  watch  the  bubble.  If  it  has  a  tendency  to  move  towards 
one  of  the  ends,  the  bubble  tube  will  have  to  be  moved  hori- 
zontally by  the  small  horizontal  capstan  screws  at  one  end  of 
the  tube.  In  some  instruments  the  screws  at  one  end  of  the 
bubble  tube  are  to  raise  it  vertically,  while  the  screws  at  the 
other  end  move  it  horizontally.  ^ 

110. — Adjustment  of  Wyes. — To  make  the  axis  of  the  bub- 
ble tube  and  the  line  of  collimation  perpendicular  to  the  ver- 
tical   axis,    level    accurately 
over   a   pair  of   screws   and 
then      turn      the      telescope          "**" 
180°.     If  there   is   any  dis- 
placement    of     the     bubble 
raise  or  lower  the  wyes  by 
the    capstan    screws    at   the 

end    of    the    horizontal    bar 

Fig.    56. 

and  correct  one-half'  of  the 

displacement.     Repeat  the  process  until  it  checks.     As  a  general 

check,  repeat  all  the  adjustments. 

111.— The  Radius  of  the  Bubble-Tube.— Let  TB,  Fig. .'56, 
the  tangent  to  the  interior  of  the  bubble  tube  cut  the  rod  at 
B,  distant  d  from  the  level,  say,  100  ft.  or  over,  turn  the  level- 
ing screws  until  the  bubble  travels  a  space  s  =  n  divisions  to 
some  point  E.  The  tangent  at  E  intersects .  the  rod  at  some 
point  C ;  take  the  difference  in  the  readings  of  B  and  C,  which 
gives  us  BC  (r)  in  feet,  measure  the  distance  s  the  bubble  trav- 
els, TE,  in  inches  and  reduce  to  feet.  The  two  tangents  TB 
and  EC  are  perpendicular  to  the  radii  consequently. 


108 


SL'Rl'EYOR'S    HAXD    BOOK. 


Angle  TOE  =  Angle   BKC  =  9 
As  the  angle  9  in  the  sectors  is  very  small  we  have 

TO  :  TE  ::  KB  :  BC 
or  R  :  s  ::  d  :  r 

*  =  T (16) 

Where  R=TO,  the  radius  of  the  bubble  tube. 

Now,  KB  is  not  exactly  equal  to  TB,  but  when  TB  is  loo 
ft.,  KB  will  be  something  like  99  ft.  11  ins.,  so  they  n  im- 
practical ly  equal. 

To  find  the  angular  value  of  one  space  on  the  bubble  tube. 
note  how  many  spaces  n  the  bubble  travels  in  the  first  opera- 
tion. In  fbE  we  have 

57  3  x  s 

R 
By  division 

H  _  573  X  s 
n  =         nR 

After  finding  one  angular  division  of  the  bubble  tube,  or 
better  the  angle  subtended  between  two  special  marks,  we  can 
use  the  level  for  measuring  distances  across  swamps,  rivers. 
etc.  Thus,,  bring  the  end  of  the  bubble  to  one  of  the  end  marks 
and  locate  the  flag  on  the  level,  and  have  the  rod  read,  shift 
the  bubble  until  the  end  reaches  the  other  mark  and  read  the 
rod  again,  take  the  difference  in  the  rod  readings  arid  call  this 
r,  the  angular  division  of  the  shift  is  O;  then  in  the  triangle 
BKC  we  have 


9   = 


Ffg.   57. 


PROBLEM  64.—  An  18-in.  Gurley  level 
gave  the  following  results  :  Distance  (d) 
=  100  ft.,  rod  reading  (r;=  0.071  ft.,  shift 
of  bubble  =  0.7  in.,  corresponding  to  seven 
divisions  on  the  bubble  tube  scale.  Find 
radius  of  bubble  tube  and  the  angle  sub- 
tended by  one  division  of  the  scale. 

PROBLEM  65.  —  If  one  angular  division 
°f  tne  bubble  tube  scale  subtends  an  angle 


LEVELING.  109 

of  21"  at  center  of  bubble  tube  circle,  find  the  distance  when  dif- 
ference of  rod  readings  was  1.28  ft,  when  the  bubble  was  shifted 
five  divisions. 

112.  Curvature  of  Earth.—  Let  AB,  Fig.  57,  be.  a  horizon- 
tal line  of  sight,  ACR  the  surface  of  the  earth.  Let  distance 
AB  =  D,  BC^c  and  radius  of  earth  r. 

In  right  triangle  OAB, 

OB*=  OA*  +~AB* 


Now  the  term  c1  is  very  small  in  comparison  with  D*  and  can 
be  omitted  without  sensible  error. 

7~)2 
/.  c  =  —  —  ,  nearly 

If  we  wish  the  correction  in  feet  while  D  is  in  miles,  we  get 


If  D  =  l  mile,  c  =  2/3  of  1  ft.  =  8  ins. 

If  D  =  2  miles,  c  =  32  ins. 

If  D  =  3  miles,  c  =  72  ins. 

Effect  of  Refraction.  —  Refraction  has  a  tendency  to  make 
all  bodies  near  the  horizon  appear  higher  than  their  natural 
positions.  Thus  if  in  Fig.  57  the  level  is  at  A,  the  line  of  sight 
will  be  the  curved  line  AK,  the  radius  of  which  is  about  seven 
times  the  radius  of  the  earth.  In  formula  for  curvature,  r  be- 
comes lr. 

.  BK    _S_     »L 

~~    2(7r)    "~l4r 
If  r  is  in  feet  while  D  is  in  miles, 

6280<5280       2D^ 


_ 
BK~  14  X  3926  X  5280""  21 


If  D  =  l  mile,  BD  =  2/21  ft.  =  1.14  ins. 

If  D  =  "%  mile,  BK  =  0.07  in. 

If  D  =  3.2o  miles,  BK  =  l  ft.,  i.  e.,  under  ordinary  condi- 
tions of  atmosphere  all  points  3^4  miles  from  the  observer  ap- 
pear 1  ft.  higher  than  their  natural  positions. 


110 


SURVEYOR'S    HAND    BOOK. 


113. — Vertical  Curves. — If  two  grades  meet  at  a  summit 
B,  Fig.  08,  it  becomes  necessary  to  round  off  this  summit  by 
uniting  the  two  grades  by  a  curve  tangent  to  each.  The 
simplest  vertical  curve  that  can  be  adopted  for  this  purpose  is 
a  common  'parabola  that  touches  the  grade  lines  at  A  and  C 
where  the  horizontal  distance  AK  =  KE,  and  AM  =  MC.  Hence 
BM  is  a  diameter  of  the  parabola  of  which  BA  and  BC  are 
tangents.  Then 


Fig.    58. 
PQ  :  BV  =  AT-  :  AK*    .  • .    PQ  =  BV  x  ~ 


PQ  :  DC  =  AT*  :  AE*    ,' .     PQ  =  DC  ) 

Let  g  =  grade  of  AB,  rise  per  station,  g'  =  grade  of  BC,  fall 
per  station,  and  ;i  =  number  of  stations  in  AB  and  BC.  Now 
BK  =  ng. 

Draw  #F  parallel  to  horizontal  line  AE 


But    FC  = 


DC  =  DF  +  FC  = 


But  PQ  -  DC 


AT* 
-= 


Now,  AE  '=•  In,  and  if  AT  one  station  = 


PQ  is   the   change  of  grade  for  the  first  station.     Let  this 
change  =  a 


•''  a=     4M 
Change  for  2nd  station  = 

Change  for  3rd  station  = 


•  (21) 


LEVELING.  Ill 

Example:  Given  g  =  1.0,  g'  =  0.8,  w  =  3,  elevation  of  B  = 
70.8',  find  the  elevation  of  different  points  on  the  curve. 

Elevation  of  R,  P,  and  A  are  75.8,  74.8,  73.8  respectively,  and 
the  decrease  in  grade  (or  elevation)  to  bring  road-bed  to 
curve  at  points  P,  and  R,  and  B  are  .15,  4  X  .15,  9  X  .15  or  .15, 
.60,  1.36. 

Hence  the  elevations  of  points  on  the  curve  are  73.8  (74.8  — 
.15),  (75.8  — .60)   (76.8  —  1.35)  or  73.8,  74.65,  75.20,  75.45. 
Original     Change  of    Grade  on 


Station. 
,4  =  56 
57 
58 
5  =  59 
60 
61 
C  =  62 

Grade. 
73.8 
74.8 
75.8 
76.8 
76.0 
75.2 
74.4 

Grade. 
.00 
.15 
.60     ' 
1.35 
.60 
.15 
.00 

Curve. 
73.80 
74.65 
75.20 
75.45 
75.40 
75.05 
74.40 

r>9. 


PROBLEM  66.  —  If  two  grades  at  a  summit  are  1.4  and  —  1.0  and 
elevation  of  summit  is  94.6,  find  elevation  of  points  on  curve  if 
«  =  3. 

114.  Curve  in  Sag.  —  If  the  curve  occurs  at  a  sag  the 
same  formulas  will  apply  in  finding  the  change  for  each  station, 
but  we  must  remember  that  the  tangents  are  below  the  curve 
and  that  all  elevations  must  be  increased  instead  of  diminished. 
Thus  in  Fig.  59,  if  grade  of  AB  =  —  .7,  and  of  5C=.5,  eleva- 
tion of  5  =  54.8,  and  w  =  3,  we  have 


a==  4n 


4X3 


Then  we  have  the  results  as  follows: 


112 


SURVEYOR'S    HAND    BOOK, 


Station. 

A  =  22 
23 
24 

5  =  25 
26 
27 


Original  Change  of  Grade  on 

Grade.  Grade.  Curve. 

56.9  .00  56.9 

56.2  .10  56.3 


55.5 
54.8 
55.3 
,55.8 
56.3 


.40 
.90 
.40 
.10 
.00 


55.9 
55.7 
55.7 

:,:,  n 
56.3 


115.  Vertical  Circular  Curves. — If  two  tangents  AB  and 
BL  meet  at  summit  B,  Fig.  60,  a  circular  curve  can  be  used  to 
unite  the  two  grades.  Let  O  be  the  center  of  circular  curve. 
Now  g  =  grade  of  AB  or  the  amount  of  rise  of  AB  per  station, 
or  100  ft.  If  the  distance  is  measured  in  stations,  g  is  the  tangent 
of  the  angle  the  first  line,  AB,  makes  with  the  horizontal.  In 
the  rt-triangle  AOB,  angle  AOB  equals  half  of  grade  angle 
DEL.  AB  =  OAtan  AOB-'.T  =  Rtan  %  DEL,  where  AB  = 
T,  and  OA=R. 


The  angle  DBL  is  very  small,  usually  less  than  4°.     .'.  we 
can  write: 

Tan  AOB=tan 
V.  T=R 
By  geometry,  AP'=PQ(2R  +  PQ) =2R   x  PQ  +  PQ* 

2 

Now  PQ  is  so  small  in  comparison  with  2R  x  PQ  that  it  can 

A  p%         AP*  ( 
be  omitted.     .  • .  AP8  =  2R  x  PQ  or  PQ  = =  ^ 


LEVELING.  113 

But  PQ=a,  and  AP=y 
Now  T—AB=n,  number  of  stations  in  AB 


The  last  formula  is  the  same  one  we  found  for  parabolic 
curve.  The  curve  is  really  so  flat  that  it  can  be  regarded  as  a 
circle  or  parabola  without  error. 


CHAPTER  VII. 
TOPOGRAPHIC  SURVEY. 

116.  Topographic  Survey. — A  compass  or  transit  sur- 
vey will  locate  points  with  reference  to  each  other  in  a  hori- 
zontal plane.  In  other  words  such  surveys  show  the  geographic 
location  of  points  with  respect  to  each  other,  but  they  do  not 
show  how  such  points  are  situated  in  elevation  with  respect  to 
each  other.  A  topographic  survey  will  give  not  only  the  rela- 
tive position  of  points  with  respect  to  their  geographic  posi- 
tions, but  will  also  give  their  elevation  vertically.  A  glance  at  the 
map  will  show  the  positions  of  the  different  objects  in  the 
geographic  relations,  but  certain  other  data  must  be  placed  on 
thc^  maps  to  indicate  the  configuration  of  the  terrain. 


Fig.    61. 

117. — Topographic  Methods. — There  are  four  general 
methods  of  making  a  topographic  survey:  (1)  By  transit  and 
level;  (2)  by  stadia;  (3)  by  plane  table;  (4)  by  hand  level. 
The  first  method  is  costly,  laborious,  and  slow.  With  the  exer- 
cise of  care,  however,  it  is  the  most  accurate  method,  but  its 
cost  and  the  labor  required  render  its  use  almost  prohibitive  ex- 
cept for  small  tracts.  The  third  method  is  coarse  but  rapid, 
and  for  large  areas  is  by  far  the  most  practicable.  It  is  suffi- 
ciently accurate  for  geologic  purposes,  and  a  survey  by  this 
method  is  a  valuable  adjunct  to  a  more  detailed  survey  by  either 
of  the  other  methods.  It  is  useless  to  discuss  here  the  methods 
of  making  a  topographic  survey  by  the  transit  and  level,  as  the 
use  of  these  instruments  is  fully  discussed  in  the  chapters  de- 

114 


TOPOGRAPHIC    SURVEY.  115 

voted  to  their  consideration.  We  shall  in  this  chapter  consider 
the  stadia  method  only. 

118.  Stadia  Formulas.  —  The  two  stadia  wires  are  placed 
in  the  reticule  of  the  telescope  of  the  transit  above  and  below 
the  horizontal  cross-wire  and  parallel  thereto.  If  these  wires  be 
represented  by  A  and  B  in  Fig.  61,  and  lines  be  drawn  from  A 
and  B  through  the  optical  center  O  of  the  objective,  these  lines 
will  cut  the  stadia  rod  at  A'  and  B'.  The  lines  A  A'  and  BB' 
are  called  secondary  axes.  If  we  let  t  represent  AB  and  r  repre- 
sent A'B',  then  from  the  similar  triangles,  OAB  and  OA'B',  we 
have,  i:r:  :f:d. 

But  by  the  law  of  lenses, 


where  F  is  the  "principal  focal  distance."  If  parallel  rays  of 
light  impinge  on  a  lens  they  will  1  e  brought  to  a  focus  at  some 
point  l\  which  is  called  the  "Principal  Focus"  of  the  lens,  and 
the  distance  OV  is  called  the  principal  focal  distance.  This  dis- 
tance can  be  found  for  any  given  lens  by  holding  the  lens  so 
that  the  central  plane  of  the  lens  will  be  perpendicular  to  the 
sun's  rays.  The  rays  of  sunlight  will  be  brought  to  a  focus, 
which  can  be  found  by  moving  a  white  sheet  of  paper  parallel 
to  the  central  plane  of  the  lens.  If  the  sheet  of  paper  is  beyond 
the  focus  from  the  lens  the  circular  disc  of  light  will  be  fringed 
with  blue,  while  if  between  the  focus  and  lens  it  will  be  fringed 
with  red  or  yellow.  When  the  sheet  of  paper  is  at  the  focus  the 
rays  of  light  will  be  concentrated  into  a  very  small  circular  disc 
of  intense  light.  To  find  F  for  the  object  glass  of  the  telescope, 
point  the  telescope  to  the  clear  sky  and  focus  on  the  cross-wires, 
and  then  measure  from  reticule  to  center  of  object  glass. 
From  the  first  of  the  above  equations,  we  have, 

1         r 

J-Td 

and  from  the  second,   . 

1        d-F 


116  SURVEYOR'S    HAND    BOOK. 

Equating  and  reducing,  we  get, 

d  =  -j  r  +  F. 

Now  the  "principal  focal  distance,"/7,  is  fixed  for  any  lens 
and  »  (the  distance  between  the  stadia  wires  on  the  reticule) 
can  be  so  adjusted  that  the  ratio  of  F  to  i  will  be  made  an> 
value  desired.  From  the  last  equation  we  have, 


But  (d  —  F)  is  the  distance  from  the  "principal  focus"  V  tc 
the  stadia  rod,  and  as  F  ~=~  /  is  constant,  we  see  that,  in  reality 
the  distance  from  the  'principal  focus  to  the  stadia  rod  varies 
directly  as  the  intercept  r  on  the  stadia  rod. 

If  we  wish  to  obtain  the  distance  D  from  the  center  of  in- 
strument to  the  rod,  we  have, 

D  =  d  +  c  =  j  r  +  F  +  c,  .........  ....  (23) 

where  c  is  the  horizontal  distance  from  center  of  objective  tc 
plumb-bob. 

In  the  majority  of  transits  the  distance  F  +  c  varies  fron 
.80  to  1.25  and  1.00  can  be  assumed  as  a  fair  average  withoui 
sensible  error. 

119.  Wire  Interval.  —  To  fix  the  stadia  wires  in  a  transit 
we  must  first  find  F,  and  then  decide  on  some  distance  from  the 
rod  to  the  principal  focus,  say  400  ft.  After  this  has  been  dom 
we  focus  on  the  rod,  then  measure  the  principal  focal  distance 
from  the  lens  of  the  objective,  which  establishes  the  principa 
focus  in  the  line  of  sight,  and  from  this  distance  we  measure 
the  400  ft.  and  set  up  the  rod  exactly  at  the  end  of  this  400  ft 
Or  we  can  measure  from  objective  to  the  rod  400  ft.  plus  the 
principal  focal  distance.  We  now  adjust  the  stadia  wires  so  tha 
while  one  of  them  (the  lower)  reads  2.00  the  upper  will  reac 
6.00,  the  difference  being  4.00.  Then, 

F  F 

400  =      4    .-.         =  100. 


TOPOGRAPHIC    SURREY. 


117 


If  the  wires  are  fixed,  find  F,  c,  D  and  r  for  a  given  reading, 
then 

i  =  Fr-h  (D  —  F  —  c). 

120. — Inclined  Sights. — If  the  line  of  sights  OC  is  in- 
clined to  the  horizon  at  an  angle  v,  as  in  Fig.  62,  we  shall  for 
the  purpose  of  mapping  have  to  find  the  horizontal  distance  O£ 
and  the  vertical  distance  CE:  The  rod  AB  is  always  held  ver- 
tically. The  lines  of  sight  as  determined  by  the  stadia  wires 
are  OA  and  OB.  Draw  A'B'  perpendicular  to  OC,  the  line  of 
sight  as  determined  by  the  cross-wires,  and  let  A'B''=r'.  The 
angle  BCB'  =  v  and  the  angles  at  A'  and  B'  differ  so  slightly 
from  a  right  angle  that  for  all  practical  purposes  we  can  assume 
them  equal  to  90. 


y 


Fig.    62. 

•  '•   B'C=-BC  cos.  v. 
2B'C  =  2BC  cos.  v. 
or  r'  =  r  cos.'  v. 

But  OC=^r'  +  F  +  c 

F 
=^7  r  c0s.  v  +  (F  +  c) 


Then  D=OE=OC  cos. 


r  cos.2  v+  (F  +  c)cos.  v. 


H—CE=OC  sin.  v—~  r  sin.  v  cos.  v 


sn.  v 


118         SURVEYOR'S  HAND  BOOK. 

=  r  sin-  2v  +  (F  +  c)  sin.  v 


.'.D  =  K  cos.''  v+(F  +  c)   cos.  r 
H  =  Vfc  K  sin.  2v  +  (F  +  c)  sin.  v  - 

Now  the  last  terms  in  the  formulas  for  D  and  H  arc  insig- 
nificant in  comparison  with  the  first  term  and  unless  refined  ac- 
curacy is  required  these  terms  can  be  omitted. 

If  F  +  <r  =  1.00,  and  if  F-M=MK).  r  =  5.40,  and  t'=6°  25', 
we  have,  £>  =  540  X  .9875  +  (1  X  .9937)  =534.24. 

//  =  HX540X.222  +  (1  X.1118)=60.0B. 

If  the  last  terms  are  omitted  we  have  D  =  533.25  and  //  =  :>'.'  '"I. 
the  errors  being  1  in  538  and  1  in  537  respectively.  For  ordinary 
maps  one-fiftieth  of  an  inch  is  about  as  fine  as  we  can  indicate 
on  the  drawing  paper.  Thus,  if  we  adopt  a  scale  of  1  in.  equals 
10  ft.,  or  one-tenth  of  an  inch  to  the  foot,  the  distance  (D) 
above  will  be  represented  by  a  line  53.4  ins.  But  if  we  adopt  a 
scale  of  1  in.  equal  to  1<X>  ft.,  which  is  the  usual  scale  in  railway 
topography,  we  would  have,  D  =  5.34  ins.  and  the  error  com- 
mitted by  the  omission  of  the  last  term  in  the  formula  for  dis- 
tance would  be  one  hundredth  part  of  an  inch. 

121.  Stadia  Rod.—  The  essentials  of  a  good  stadia  rod 
are  that  it  should  be  clearly,  accurately  and  distinctly  graduated 
and  that  the  graduations  should  be  sufficiently  clear  to  be  read 
to  the  extreme  limits  of  its  longest  range.  There  are  many 
special  rods  on  the  market,  each  possessing  special  merits  in  the 
opinion  of  the  designer,  but  the  Philadelphia  rod  can  be  used 
while  the  marks  are  new  and  clear  cut.  Fig.  63  shows  one 
form  of  stadia  rod  that  is  extensively  used.  It  is  3.5  ins.  wide, 
%  in.  thick  in  the  body  where  the  graduations  are  placed,  and 
%  in.  thick  on  the  edges.  The  rod  is  made  of  straight  grained 
wood,  is  12  ft.  long  over  all  and  is  hinged  in  the  middle  so  that 
it  can  be  folded  for  convenient  transport.  The  raised  flango 
(Hxl/16  in.)  afford  excellent  and  effective  protection  to  the 


TOPOGRAPHIC    Sl'Rl'EY.  119 


graduation-;  The  foot  marks  are  indicated  in  red 
figures.  J/J-")  to  0.75  in.,  while  the  tenths  are  indi- 
cated by  black  figures,  0.75  in.  high  by  0.5  in.  width. 
The  space  is  divided  in  alternate  black  and  white 
strips  one-hundredth  of  a  foot  in  width.  Each  red 
fignre  is  opposite  a  black  strip  2.5  ins.  long,  and  the 
liiiure  refers  to  the  toh  edge  of  the  strip  and  indi- 
cates its  distance  from  the  bottom  of  the  rod.  In 
the  same  way  each  black  figure  is  opposite  a  black 
strip  of  sanu  width  but  only  1.25  in.  in  length,  the 
black  figures  indicating  the  distance  in  length  of  a 
foot  of  the  top  of  its  strip  from  the  top  of  the  strip 
through  the  red  figure  below.  The  space  between 
the  black  figures  (the  top  through  the  black  lines) 
is  divided  into  ten  equal  spaces  alternately  painted 
black,  while  the  white  background  forms  another 
strip  of  the  same  width.  If  the  wire  reads  between 
the  Fed  3  and  4,  between  the  black  G  and  7,  and  is 
at  the  top  of  the  third  black  strip,  the  reading  is 
3. HO.  It  is  well  to  remember  that  the  top  of  the 
short  black  strips  (about  %  in.  long)  indicate  even 
hundredths,  i.  e.,  .02,  .04,  .06,  etc.,  while  the  bot- 
tom of  the  black  strip  indicates  the  odd  hundredths. 
These  remarks  apply  (except  as  to  lengths  of  the 
black  strips)  to  the  Philadelphia  rod,  which  for  dis- 
tances under  fioO  ft.  forms  an  excellent  stadia  rod. 
122.  Field  Work. —  When  it  is  desired  to  make 
a  topographic  survey  of  a  certain  district  by  the 
stadia  method,  certain  base  lines  or  lines  of  refer- 
ence are  adopted  as  a  basis  to  tie  into.  If  the  dis- 
trict has  been  surveyed  by  triangulation,  the  trian- 
gulation  stations  form  the  points  from  which  the 
survey  proceeds.  The  transit  is  set  up  over  one  of 
these  triangulation  stations  and  sighted  to  an- 
other station  of  the  triangulation  survey.  The 
azimuth  of  this  line  has  been  previously  deter- 
mined and  the  transit  can  be  adjusted  by  upper  Fig.  63. 


120 


SURVEYOR'S    HAND    BOOK. 


motion  so  that  the  zeros  of  the  verniers  point  north  and  south. 
When  the  transit  has  been  set  and  adjusted  so  that  the  zeros  will 
mark  out  the  true  meridian,  the  instrument  man  can  send  his 
rod  man  to  certain  strategic  points  in  the  terrain.  The  distance, 
azimuth  and  angle  of  elevation  must  be  read  and  recorded. 

To  obtain  the  distance  the  lowest  stadia  wire  is  brought 
preferably  on  some  even  foot-mark,  as  the  1  or  2,  and  the  upper 
wire  is  then  read  7.42.  The  difference  is  5.42  and  the  distance  by 
stadia  542  ft.  To  obtain  the  angle  of  elevation,  the  middle  cross- 
wire  must  be  brought  on  the  mark  on  the  rod  that  indicates  the 


height  of  the  center  of  the  horizontal  axis  of  the  telescope.  It  is 
necessary  for  the  transit  man  at  every  set  up  to  take  the  height 
of  the  telescope  above  the  surface  under  the  plumb-bob.  The 
azimuth  is  read  from  the  south  by  west,  north,  east  and  on  to 
south  again. 

The  primary  triangulation  stations  are  indicated  by  the  sym- 
bol A,  while  the  stadia  stations  are  marked  [T]with  a  number 
following  to  define  it,  as  [V]  3,  [][]  7,  etc.  If  there  has  been  no 
triangulation  survey  the  topographic  survey  proceeds  from  the 
same  local  point  to  which  the  stadia  stations  are  connected  or 


TOPOGRAPHIC  <  SURVEY, 


121 


"tied  in."  Other  points  are  variously  described  in  the  "object" 
column  as  "house,"  "tree,"  "cor.  fence."  If  a  reading  is  taken 
simply  for  a  contour  point  it  is  marked  C.  P. 

Smith,  Instrument. 

Henry,  Recorder. 

Fox,  Rod. 


Oct.  14,  1907. 
At  [J  J  Ht.  of  Inst.  =  5'.l 
Object.                   Azimuth.  T 

CP    229°  15' 

)istancc 
Ft. 
99 
206 
332 
370 
387 
281 
294 
181 
81 
163 
401 
754 
Mean  : 
401 
90 
171 
204 
445 
280 
78 
150 
250 
331 
Mean 
755 
227 
250 
294 
250 
103 
175 
331 

Elevation  =  500'.0( 
>.  Vert.  Angle.  Diff.  of  El. 

—1—52'          —  3'.2 
0—48'              2'.9 
1_24'             8'.1 
1—  4'              G'.9 
1—40'            ll'.O 
1_50'              g'.o 
2—  8'            10'.9 
0—56'             3'.0 
—8—  38'        —  12'0 
0_42'              2'.0 
5_58'            41'.40 
1—47'            23'.45 
=  41'.52                        541' 
_5_59'        —  41'.58 

—4—32'        —  13'"5 
—3—50'        —  17'.0 
_2_  50'        —  22'.7 
_o—22'        —  20'.  1 
_8—58'        —  12'.0 
_6—  30'        —  10'.9 

—5—18'        _  30''5 
=  23'.69                       523' 
—1—49'        —  23'.93 
_1_44'          _f/.9 
_0_l38'          —  2'.8 
__!_  8'          —  5'.8 

—3—42'        —  10'is 
_1_26'         —  4'.4 

1O/V                          O  '"  7 
.""^•DU                    O    4 

) 

El. 

496'.8 
502'.  9 
508'.  1 

5oe;.9 

509''.i) 
510'.9 
503'.0 

488;..0 

219°  12' 

« 

210°  00' 

u 

228°  45' 

a 

218°    5' 

(C 

254°  10' 

1C 

283°  30' 

it 

290°  -  8' 

".     

320°  15' 
64°  38' 
.157°  17' 

[T]  3  

246°  51' 

.52 

At  [T]  2  Ht. 

of  Inst.  =  4'.8 
387°  17' 

CP 

229    12' 

535'.0 
528'.0 
523'.9 
5.1S'.8 
515'.4 
529'.5 
524'.0 
513'.9 

.09 

510'.8 
520'.9 
517'.9 
509'.  1 
513'.2 
519'.3 
532'.4 

244°  30' 

« 

..'..  .252°  30' 

it 

266°  00' 

a 

269°  38' 

u 

297°  -18' 

u 

18°  -  5' 

it 

316°  15' 

At  [T]'  3  Ht. 
HI 

356°  -10' 
66°  51' 

x  
CP 

.  .  66°  30' 

1C 

26°  47' 
00°  35' 

(C 

....  97°  .  8' 
.  .  .  .  133°  20' 

M 

,...162°  40' 

U 

..117°    5' 

122 


SURVEYOR'S    HAND    BOOK. 


123.     Reduction     Methods.— The  formula    for    finding    the 
elevation  of  a  point  above  the  instrument, 

//  =  inclined  distance   X   sin.  v. 

When  v  is  less  than  6°,  we  can  find  //  readily  by  the  application 
of  the  57.3  rule.  But  to  save  time  several  labor-saving  devices 
have  been  invented.  Two  of  these  make  use  of  the  principle  of 


COX'S   STADIA   COMPUTER. 


Directions  for  Use. 

Pet  the  arrow,  ir 
reading  of  the  rod 

Opposite  the  vertical  angle  of  the  transit  telescope  find 
the  Difference  of  Elevation,  and  opposite  the  same  angle 


rked  zero  on  the  disc,  oppos 
in  the  outer  scale. 


the  Distance  t 


>le  rind  the  II. 


EXAMPLE: 

•al  angle  12 '  30'.  reading  of  the  Rod  537  feet.    Set  the 
o  of  the  disc  opposite  W7,  and  opposite  12'  30-  of 
the  scale  at  the  left  rend  113)  feet  Difference  of 
Klevation,  and  opposite  12"  30',  of  the  Scale 
at  the  right  read   512  feel  Distance. 


Copyr.gh 


Detigncd  by  Wm.  Cox. 


Fig.    65. 

the  slide  rule,  Colby's -Slide  Rule,  which  can  be  obtained  from 
the  leading  dealers  in  drawing  supplies  and  mathematical  instru- 
ments, and  Cox's  "Stadia  Computer,"  manufactured  by  W.  & 
L.  E.  Gurley,  Troy,  N.  Y.  This  "Stadia  Computer,"  Fig.  65,  is 
simply  a  circular  slide  rule  about  15  ins.  in  effective  length.  It 
consists  of  a  mounted  card  board,  6%x6}4  ins.,  upon  which  scale 


TOPOGRAPHIC    SURVEY. 


is  laid  off  the  logarithm  of  numbers  from  1  to  1,000  on  the  cir- 
cumference of  a  circle  5  ins.  in  diameter.  Mounted  on  this  scale 
is  a  circular  disc  concentric  with  the  o-in.  circle  on  the  limb,  on 
which  is  laid  off  the  logarithm  of  the  sines  of  angles  from  3'  up 
to  45°.  To  find  the  difference  of  elevation  for 
any  distance  and  angle  of  elevation,  turn  the 
moving  disc  till  the  zero  of  the  disc  is  oppo- 
site the  required  distance.  Hold  the  disc  in 
this  position  and  opposite  the  given  angle  of 
the  disc  read  the  number  on  the  limb.  This 
is  the  required  difference  in  height.  The 
horizontal  distance  is  read  opposite  the  angle 
in  the  space  marked  "Hor.  Distance." 

EXAMPLE:  Given  distance  —  480,  angle  of 
elevation  =  5°  10',  find  the  difference  of  ele- 
vation. Turn  the  disc  till  the  zero  is  oppo- 
site 480  on  the  limb  and  then  opposite  5°  10' 
on  the  disc  read  43  ft.  The  whole  computer 
can  be  carried  in  the  coat  pocket  and  its  con- 
venient size  makes  it  a  very  effective  calcula- 
tor. No  correction  for  horizontal  distance  is 
necessary  for  this  angle  of  elevation. 

124,  Colby's  Slide  Rule.— Colby's  Slide 
Rule  as  shown  in  Fig.  .66  consists  of  a  base 
piece  of  trapezoidal  cross  section  on  which  is 
laid  off  the  logarithm  of  the  numbers  repre- 
senting the  distance  read  by  the  stadia,  and  a 
sliding  runner  on  which  is  laid  off  the  angles 
of  elevation  to  18°  30'.  On  the  sliding  run- 
ner is  a  mark  labeled  "same  unit  index," 
which  can  be  seen  on  the  right  on  the  run- 
ner above  the  space  between  the  numbers  3 
and  4.  To  find  the  vertical  distance  between 


i!  I 


Fig. 


the  instrument  and  rod,  set  the  mark  under  "same  unit  index" 
to  agree  with  the  distance  read  by  the  stadia,  and  then  opposite 
the  angle  of  elevation  on  the  slide  read  the  vertical  distance 
on  the  log  scale  below. 


124  SURVEYOR'S    HAND    BOOK. 

EXAMPLE:  Given  distance  600  and  angle  of  elevation  3°  10', 
to  find  the  difference  of  elevation.  Set  index  on  slide  opposite 
<>00  on  log  scale,  and  opposite  3°  10'  on  the  slide,  read  33.1  on 
log  scale,  which  is  the  difference  of  elevation. 

125.  Usual  Approximations. — The  cosine  of  all  angles 
less  than  18°  is  greater  than  0.95  and  we  may  assume  F+f  — 1 
and  (F  +  c)  cos  £>  — .95.  Now,  if  the  horizontal  distances  are 
to  be  read  to  the  nearest  tenth  of  a  foot,  we  can  assume  (F  +  c) 
cos  v  =  1.  The  following  approximations  may  be  made : 

(1)     If  the  last  term  =1  and  D  =  K.  in  the  formula, 
D  =  K  cos.\<  +  (F  +  c)  cos.v,  we  have 
D  =  K  cos.  V  +  1 

or  K  =  K  cos.\<  +  1 

Cos-v 

Now   if  #  =  200, 

•_)( M )  =  200   Cos.*r  +  1  • ' •  7'  =  4°04' 
If  /C  — 700,  v  =  2°W 

Thus,  if  the  angle  of  elevation  is  2°  10'  and  the  inclined  distance 
700,  we  can  omit  the  last  term  and  make  the  horizontal  dis- 
tance equal  to  the  inclined.  The  two  approximations  or  as- 
sumptions balance  each  other.  Check : 

D  =  700'f0J22°  W  +  cos2°  W 
=  700  X  .1)086  +  .9993 
=  699.02 +  .9993  =  700.02 

For  an  agle  of  elevation  of  2°  10'  and  a  distance  of  less  than 
7oo  (say,  500)  we  have 

D  —  500  X  .9986  +.9993  =  500.3 

For  all  distances  less  than  700  and  a  given  angle  of  2°  10'  the 
horizontal  distance  D  will  be  greater  than  A',  but  the  error  is  less 
than  1  foot.  For  all  distances  above  700  the  horizontal  distance 
(D)  is  less  than  K,  but  the  error  is  less  than  one  foot  when  K  is 
les  than  1,400'.  The  following  table  gives  the  values  of  v  for 
certain  distances  when  D==K: 


TOPOGRAPHIC    SURVEY.  125 

K                  Angles                     K  Angle  v 

100                   5°  44'                      700  2°  10' 

200                   4°  04'                      800  2°  02' 

300                   3°  20'                      900  1°55' 

400                   2'°  52'                   1,000  1°  49' 

500                   2°  34'                   1,100  1°44' 

600                  2°  20'                  1,200  1°40' 

(2)  When  D  is  1'  less  than  K,  \.  e.,  for  error  of  1  ft.  when 
the  last  term  =  1'.   we  have, 

D  =  K  —  1; 
D  =  K  cos.*v  +  1 ; 
or  AT—  1  =  K  cos*v  +  1, 

2 
.-.  cos.'zv=  1  —  jp: 

Solving   for  the   different  values  of  K,  we  can  fill  out  the  fol- 
lowing table : 

K                  Angles/                      K  Angle  v 
100                  8°  08'                     700  3°  04' 
200                   5°  44'                      800  2°  52' 
300                  4°  41'                     900  2°  42' 
400                   4°  03'                   1,000  2°  34' 
500                   3°  38'                   1,100  2°  27' 
600                  3°  20'                   1,200  2°  20' 
For  any  angle  given  in  table  and  distance  less  than  the  cor- 
responding value  of  K,  the  error  in  D  will  be  less  than  1  ft. 

(3)  When  last  tgrin—-!*  and  there  is  a  total  error  of  1  per 
cent  in  horizontal  distance,  we  have  D  =  .99K, 

D  =  K  cos*v  +  1 

or  .99AT  =  K  cos.*v  +  I     .' .     cos*v  =  .99  —  J? 

A 

This   formula  gives   the   following : 

K                  Angles                      A'  Angle  v 

100                  8°  08'                     700  6°  08' 

200                   7°  02'                      800  6°  05' 

300                   6°  38'                      900  6°  03' 

400                   6°  25'                   1,000  6°  OK 

500                   6°  17'                   1,100  6°  00' 

600                  6°  12'                  1,200  5°  59' 
To  find  D  from  table,  subtract  1  per  cent. 


126 


SURVEYOR'S    HAND    BOOK. 


EXAMPLE:     If   A'  =  800,    we  get   D  =  800  —  8  =792. 
(4)     If  the  last  term  be  omitted  and  there  is  an  error  of  1 
per  cent,  i.  e,,  if  there  is  a  total  error  of  1  per  cent  minus  1  ft., 
or   if  £>  =  .99A'  +  1,  we  get, 

D  =  K  cos.\>  +  \ 
But  D  =  .MK  +  1 
.99  K  +  1  =  K  cos.-v  +  1 
.•.cos.sv  =  .99.'.v  =  5°  44' 

That  is,  if  the  angle  of  elevation  be  5°  44'.  the  horizontal  dis- 
tance (D)  will  be  less  than  the  inclined  (K)  by  1  per  cent  of 
K  less  1'  or 

K 
.  • .  Error  =          -  1 


B 


L 


p 
Fig. 


D  =  K  —  Error. 

126.  Topography  by  Hand- 
Level. — The  hand  level  can  be 
used  economically  to  obtain  the 
data  for  a  topographic  map  of  any- 
small  area.  A  base  line  should  be 
adopted  from  which  the  survey 
proceeds,  and  lines  perpendicular 
to  this  base  line  should  be  drawn 
at  known  intervals.  Thus,  if  in 
Fig.  67,  A  BCD  represents  a 
section  of  area,  adopt  a  base 
line  PQ  and  at  points  P, 
1,  2,  3,  and  Q  locate  lines  normal  to  PQ.  These  lines  should 
be  marked  out  by  stakes  so  they  can  be  easily  followed.  In 
order  to  leave  all  elevations  positive,  assume  some  datum  be- 
low the  lowest  point  and  refer  the  elevations  of  all  points  to 
this  datum.  Begin  at  some  point  as  P  and  find  the  elevation 
of  points  along  this  line.  The  notes  should  be  kept  so  the 
height  of  any  point  will  appear  as  the  numerator  of  a  frac- 
tion, while  its  distance  out  from  base,  line  will  appear  as  the 
denominator.  The  height  of  the  eye  should  first  be  deter- 
mined and  rod  readings  should  be  taken  at  a  sufficient  number 
of  points  to  determine  the  configuration  of  the  landscape.  The 


TOPOGRAPHIC    SURVEY. 


127 


bench  mark  should  be  located  somewhere  below  the  point  C, 
and  from  this  the  levelman  makes  his  observation  on  the  rod 
held  on  some  point  in  line  DC.  The  difference  of  the  rod  read- 
ing and  height  of  eye  will  give  the  elevation  of  the  point  of 
rod  above  the  observer.  Thus,  if 

h  =  height  of  eye  of  observer, 
r— rod  reading,  then, 

h  —  r  —  elevation  of  rodman  above  observer. 
If  h  —  r  is  negative,  the  rodman  is  below  the  observer. 
The   following  notes   were  taken   on   a   hand-level   survey  of 
a  rectangular  area : 


Line         Left  of  PQ 


Base 
Line 


Right  of  PQ 


33     28 

DC.  . 

200     100 

34   2.9   25 

1  

200  100   50 

35   29   26 

2 

200  100  50 

36   31   27 

3  

200  100  50 

37   33   31 

24 

19 

20 

15 

11 

14 

0 

100 

150 

200 

250 

300 

23 

18 

15 

14 

16 

20 

0 

100 

150 

200 

250 

300 

24 

22 

21 

22 

23 

25 

0 

100 

150 

200 

250 

300 

26 

25 

26 

27 

28 

30 

0 

100 

150 

200~ 

250 

300 

29 

30 

31 

32 

33 

35 

IT 

100 

150 

200 

250 

300 

200      100     50 

BIBLIOGRAPHY. — "A  Manual  of  Topographic  Methods/'  by 
Henry  Gannett.  This  work  is  published  by  the  United  States 
Geological  Survey  and  its  title  indicates-  its  scope,  as  it  deals 
only  with  the  theory  of  topography,  but  gives  also  the  illustrated 
methods  as  practiced  by  the  engineers  of  the  Survey,  the  most 
expert  topographers  in  the  world. 

"Topographic  Surveying,"  by  Herbert  M.  Wilson,  910  pages. 
Fully  illustrated,  having  18  engraved  colored  plates,  181  half- 
tone plates  and  many  smaller  figures.  In  addition  to  the  ex- 
cellent illustrations  of  the  best  executed  topography,  the  field 
instruments  and  other  equipments  for  field  parties  are  described 
and  the  methods  explained. 


128  SURVEYOR'S    HAXD    BOOK. 

"Elevation  and  Stadia  Tables,"  by  Arthur  P.  Davis.  These 
tables  are  for  use -in  reducing  inclined  sights  to  the  horizontal 
and  for  rinding  the  difference  of  elevation  of  observer  and  points. 


CHAPTER  VIII. 
RAILROAD   SURVEYING. 

127.  Railroad  Surveying. — By  railroad  surveying  is  meant 
the  use  of  transit  and  level  in  selecting  and  locating  the  center 
lines  of  the  track.     The  location  of  the  straight  sections  of  the 
track  is  a  matter  easily  accomplished,  but  it  becomes  necessary 
to  unite  two   straight  sections  of  track  that  intersect  at  a  defi- 
nite angle.     That  a  train  may  pass  gently  from  one  straight  line 
to    another,   making  an   angle   with   the   first,    the  two    must   be 
connected    with   each   other  by  an   intermediate   curve   to   which 
each    straight    line    is   tangent.     On   account   of   the    ease   of   lo- 
cation circular  curves  are  universally  used  to  connect  two  straight 
sections    of   track    whose    directions    are    not    the    same.     These 
straight  portions  may  be  joined  by  a  curve 

of  either  great  or  small  radius,  depending 
upon  the  character  of  the  ground.  The 
magnitude  of  the  curve  is  defined  by  the 
size  of  the  angle  that  a  100-ft.  chord  sub- 
tends at  the  center  of  the  circle.  Thus,  in 
a  4°  curve  the  100-ft.  chord  subtends  an 
angle  of  4°  at  the  center  of  the  circle.  In 
a  3°  curve,  3°  at  the  center,  etc. 

128.  Degree  Formula.— In   Fig.  68  let  AEB  be  a  circular 
arc  with  O  as  center,  and  let  ^5  —  100  ft.  and  angle  AOB  =  1). 
Then,  if  OC  is  perpendicular  to  AB, 

AC  =  CB  =  50  ft.  and  AOC  =  BOC  =  £  D 

AC 


Now, 


Sin.  AOC 


AO 


.  * .  Sin.  W  =  -£    (24) 

129.  General  Formula. — In  any  curve  AKB,  Fig.  69,  let 
AB  —  chord  c;  AP'==-  tangent  T,  AO^=-  radius  R,  FK==mid. 
ordinate  M,  PK  =  External  Et  I  =  angle  of  intersection  GPB  = 
^05. 

129 


130 


SCRl'ID'OR'S 


BOOK. 


\\\   the   right  triangle  AOP 

AP 

Tan.  AOP  = 


•  .  Tan. 


.-.  T  =  R  Tan.  £ 
In  rt.  triangle  AFO, 


AF 

•>in.  AOF  =  T7S 


.  ' .  sn.   $1  = 


2R 
2R  sin. 


In  rt.  triangle  AFK, 


Tan.FAK  =  ^~F» 

.-.r«n:i/~g 

.  '  .  M  =  $c  tan.  \  I . 
In  the  triangle  A KP, 

PK  AP 


(27) 


sin.  PAK 


PK 


If  7  is  known  and  it  is  desired  to  pass  a  curve  through  some 
point  on  the  bisector  PO,  we  measure  the  distance  PK  —  E,  and 
from  formula  (28)  calculate  7'.  Then  find  R  from  (25)  and  D 
from  (24). 

130.  To  Lay  Out  Curve.— Let  QA,  Fig.  70,  be  a  straight 
line  or  tangent  from  which  a  curve  turns  off  at  A.  The  point 
A  where  the  curve  begins  is  called  the  "Point  of  Curve"  or 
P.  C,  while  the  point  B,  where  we  pass  from  the  curve  to  the 
new  tangent  is  called  the  ''Point  of  Tangent,"  or  P.  T.  To 
lav  out  curve,  set  up  the  transit  over  the  station  at  A,  level  up 


RAILROAD   SURVEYING.  131 

and  back  sight  on  a  tack  point  in  tangent  line  AQ.  Revolve 
the  telescope  and  turn  off  the  angle  of  deflection,  which  is  half 
the  degree  of  curve.  The  rear  chaimr.an  holds  end  of  the  chain 
(the  zero  of  chain  or  tape)  on  the  tack  point  at  A,  and  the  head 
chainman  swings  his  end  of  the  chain  around  until  the  transit- 
man  catches  the  flag  pole  in  field  of  view.  The  flag  pole  is 
brought  accurately  to  coincide  with  the  line  of  sight  and  when 
the  head  chainman  has  the  chain  or  tape  straight,  a  peg  is 
driven  at  the  point  /,  which  is  a  point  on  the  curve.  The  chain- 
men  now  advance  until  the  rear  chainman  reaches  point  1,  the 
transitman,  in  the  meantime,  having  set  the  deflection  angle 
again.  The  rear  chainman  holds  the  end  of  chain  or  tape  on 


Q 

Fig.    70. 

point  1,  while  the  head  chainman  is  ranged  in  the  line  of  sight 
A'2.      When   the   chain    is    straight   and    the   flag   pole    is   in    the 
line  of  sight,  a  peg  is  driven  at  this  point  2.     In  the  same  way 
the  other  full  station  points  on  the  curve  are  located. 
Example.— Given  D=23  30'  and  /=153  54' 
50  50 

Now>  R=,^r^=^r^=2292/  -° 

1.V54'    X  100 
Length  ot  curve=    — rp  on/ —      =b3o  le-jt.  - 

The  total  angle  to  deflect  will  be  %  /  or  7°' 57'.  The  angle 
of  deflection  is  1°  1-V  and  there  will  be  six  full  deflections  of 
1°  15'  each,  making  7°  30',  and  a  partial  deflection  of  '27',  cor- 
responding to  a  chord  of  36  ft.  The  usual  curve  is  so  flat  that 


132  SURVEYOR'S    HAND    BOOK. 

the  angle  of  deflection  for  fractions  of  100  ft.  is  proportional 
to  the  length  of  chord.  Thus,  if  the  deflection  angle  for  100  ft. 
is  1°  15',  then  the  deflection  for  3(5  ft.  should  be  M  X  1°  15'  = 
-7',  which  checks  the  result  found  by  subtraction. 

131.  Obstacles. — It   often   happens   that  some   object  will 
interfere  with   our   line   of  -sight   and   we   cannot   locate   all   the 
stations   from  the    P.    C.     Suppose  that   there   were   a   house  or 
some  other  obstruction   interfering  with   the   line   of   sight   from 
the  P.  C.  to  station  5.     In  this  case  the  transit  must  be  trans- 
ferred to  station  4,  where  it  is  set  up,  leveled  and  a  back  sight 
taken  on  the  rear  flag  at  A,  the  P.  C.     Now,  if  G4  is  a  tangent 

to  the  curve  at  4,  the  angle  G4A  =  GA4. 
Hence,  if  we  turn  the  telescope  through  an 
angle  equal  to  the  angle  GA4,  the  amount 
deflected  from  the  tangent  AP,  the  line  of 
sight  will  define  the  tangent  46".  Set  t he- 
transit  at  4,  level  up,  bring  the  verniers  to 
zero,  reverse  the  telescope  and  set  on  A. 
Plunge  the  telescope  and  set  the  vernier  to 
read  6°  15',  and  the  line  of  sight  will  de- 
fine the  line  45.  This  is  more  fully  ex- 
plained and  exemplified  in  Article  139. 

132.  Location  by   Offsets. — Let  ABC,   Fig.   71     be  a   cir- 
cular    curve     when     AB  =  BC  =  C,     and      where      OA  =  OB 
—  R.     Through  B  draw  BE  parallel  to  OA   to  cut  the  tangent 
AE  at  E.     Draw  OK  perpendicular  to  AB.     Then  the  triangles 
OAK  and  ABE  are  similar. 

.'.EB  :  AB  =  AK  :  AO. 

Now,  EB  is  called  the  offset  from  the  tangent  to  curve  or  simply 
tangent  offset. 

Let  EB=d 

.-.d  :  C=\C  :  R 

C2 
d=xi .....(29) 

IMB  =  C7=100, 
5000 


RAILROAD   SURVEYING.  133 

Let  CF  be  drawn  parallel  to  OB,  to  cut  chord  AB  produced  at  F, 
and  let  BG  be  the  tangent  at  B,  cutting  CF  at  G.  Then  triangle 
BCG  =  BGF. 

But  BCG  =  ABll.     .'-CG  =  BE, 
But     CF  —  2  X  CG  =  2X5E  =  2  d. 

C2 

.  •  .  chord  offset  CF  —  -^r- 

If  C  =  100, 

10,000 
chord  offset  =  —  ^  -- 

The  formula  for  the  chord  offset  may  be  written 

,-£.™°°*-l.7«>  .............  (30) 

Thus,  for  a  1°  curve  the  chord  offset  is  1.75,  and  that  for  any 
other  curve  can  be  found  by  multiplying  1.75  by  the  degree  of 
the  curve*. 

133.     Middle  Ordinate.  —  In     Fig.  69  we  have  by  Geometry, 
KF  (2R—KF)=AFxFB. 

.-.M  (27?-M)=  -Cx~C  *J> 


Now  M2  is  small  in  comparison  with  R,  and  in  all  practical 
cases  can  be  omitted.  & 


134.     Approximate    Formulas.  —  We    have    established    the 
formula,  1  50 

sin.  ~9~£)=~^' 

Now  if  D  is  no  larger  than  8°  we  can  substitute  the  circular 
measure  of  the  angle  for  its  sine,  that  is 

J_      _*  '     D° 
sin.  2  D  —  2  ;57  2965 

D        _50 
'  ''2x57.2965""^ 

5729.65 
.  '  .  D  =        5 


134  SURVEYORS    HAND    BOOK. 

This  is  usually  written, 

n  _  573" 

'••.K-2ff-! , (32) 

Now  if  D=l,  R=5730  ft.     We  have  the  general  formula, 

T  =  R  tan.        =          -  tan.  U. 


C  =  2R  sin.  \l  =  '2  --  sin 


Let  /  remain  fixed  and  7\  and  d  be  the  tangent  and  chord 
for   1 -degree    curve.     Then, 

Ti  =  5730  tan.  \l 

Ci  =  2  x  5730  sin.  %I 

_"     Ti 

.-.  r—  D 


Again,  we  have, 

M  =\C  tan.  %I  —  — jj-  tan.  $1  sin. 


5730 
£  =  7"  tan.  -\I  =  --  tan.  U  tan.  \l. 


For  a  1°  curve  these  become, 

.l/i  —57  30  tan.  \l  sin.  $L 
El  =  5730  tan.  \1  tan.  \l. 


Then  for  all  curves  for-  a  fixed  I}  we  have, 

D  X  T  =  7\  =  a  constant, 

D  X  C  —  CY—  a  constant, 

D  X  M=M,  =  a  constant, 

D  X  E=Ei  =  n  constant. 

135.  Reduction  Tables.—  The  value  of  the  tangent  7\, 
the  long  chord  C\,  the  mid-ordinate  Al\,  and  the  external  J5i 
have  been  calculated  for  a  1-degree  curve,  corresponding  .to 


RAILROAD    SURVEYING. 


135 


Value  of  /  from  0  to  117°,  for  intervals  of  two  minutes.  To 
obtain  the  values  of  T,  C ,  M,  or  E,  it  is  only  necessary  to  look 
for  these  for  a  1-degree  curve  for  the  proper  I,  and  then  to  di- 
vide by  the  value  of  D. 

EXAMPLE  :     Find  T,  C,  M,  and  E,  for  a  4°  curve  when  /  =  21°. 
For  a  1-degree  curve,  we  get 

7!  =  1062.0.  CV=  2088.5,  Mi =95.95,  £1  =  97.58. 

. ' .  T  =  M.  X  1062  =  265.50, 
C=V4  X  2088.5  =  522.125, 
M=1A  X  95.95  =  23.988, 
£  —  14  X  97.58  =  24.395. 

136.  Metric  Curves. — Tn  Mexico  and  the  South  American 
countries  a  chain  or  tape  of  20  meters  is 
used  instead  of  the  JOO-ft.  tape  that  is  used 
in  the  United  States.  The  degree  of  the 
curve  is  the  angle  at  the  center  of  the 
circle  subtended  by  a  chord  of  20  meters. 
Thus,  in  Fig.  72  if  ,45  =  20  meters,  and 
.405  =  7}°,  the  number  of  degrees  in  the 
angle  D  gives  tl  2  degree  of  curve. 

Sin.AOK  =  TTA 


Fig.    72. 


Sin 


10 


//  D 


one  degree,  we  have, 

But  s^ne  30'  = 


.10 
Sin  30'  =  -F>- 


1 


2  x  57.3 
=  1146  meters. 


__!_    _  10 
*'•    114. (3  ~   R 

Now,  the  radius  of  a  1-degree  curve  for  the  foot  system  (pre- 
vailing in  the  United  States)  is  5730  ft  =  5  X  1146. 

In  the  same  way  all  the  functions  of  a  1-degree  metric  curve 
are  one-fifth  of  the  corresponding  functions  of  a  1-degree  curve 
of  the  foot  system.  Thus,  if  7  =  12°  7  =  602.2',  £  =  31.56',  C  — 
1197.9',  for  a  1-degree  foot  curve.  Then  7  =  120.4  meters,  E 
=  6.3  meters,  C  =  239.6  meters,  which  were  obtained  by  dividing 
the  former  values  of  7,  E  and  C  for  the  foot  curve  by  5. 


13G  SURVEYOR'S    HAND    BOOK. 

Again,  if  we  have  7  =  14°  30',  and  wish  to  find  T,  E,  and 
C  for  a  3°  metric  curve,  we  can  find  T,  E,  and  C  from 
the  usual  tables  for  the  foot  curve  and  divide  the  results  by 
live  times  the  degree  of  curvature  for  the  metric  system.  Thus, 
for  7  =  14°  30'  we  have  for  a  1-degree  curve  7  =  728.87,  E  = 
46.18,  C=144(>.2.  Then  for  a  metric  curve  of  3°  we  divide 
these  values  of  T,  E  and  L.  C.  by  3  X  5  =  15,  as  follows : 

T  =  Jg  (728.87)  =  48.59  meters, 
E  =  ^s  (46.18)  =  3. 08  meters, 
C  =  1*5  (1446.2)  =  96.41  meters. 

137.  Preliminary  Survey. — The  first  instrumental  survey 
on  a  projected  railway  line  is  called  the  preliminary  survey  and 
consists  in  running  a  traverse  line,  staking  the  line  out  by  means 
of  pegs  or  stakes,  which  are  driven  at  the  hundred-foot  marks, 
or  "stations,"  as  they  are  called,  or  at  fractional  parts  thereof. 
When  the  survey  is  finished  these  stakes  mark  out  a  polygonal 
traverse    or    survey.     There   may   be    two   or   more    preliminary 
surveys  between   the  same   termini,   and  a  comparison  of  these 
as  to  cost  of  construction,  revenue  to  be  derived  from  probable 
traffic,  and  operating  expenses  will  decide  the  most  advantageous 
route.     Fig.  73  is  a  double  page  illustration  of  the  form  of  notes 
used  in  the  field  in  preliminary  survey. 

138.  Location    Survey. — When    one    of    the    preliminary 
surveys  or  routes  has  been  adopted,  the  center  line  of  the  pro- 
posed   track    is    then   located.     The    different    tangents    must    be 
connected   by   curves   and   the   whole   line  must  be   surveyed  by 
transit,  running  in  the  curves  and  driving  new  stakes  or  chang- 
ing the  position  of  the  old  ones.     As  the  curve  is  shorter  than 
the  sum  of  the  two  tangents,  the  first  P.  T.  will  be  less  in  dis- 
tance from  the  beginning,  that  is,  all  stakes  after  the  first  P.  C. 
will  be  moved  forward.     Those  on  the  tangents   (from  P.  C.  to 
P.  I.  and  from  P.  I.  to  P.  T.)  will  be  moved  over  to  the  curve 
and  all  those  on  the  part  of  tangent  from  the  P.  T.  to  the  next 
P.  C.  ahead  will  be  moved  forward  so  that  the  number  of  each 
stake  will  give  its  distance  from  the  beginning  as  measured  along 
the  proposed  center  of  track.     Thus,  if  the  angle  of  intersection 
7=:]60   00'  and  we  unite  the   two  tangents  by  a  4°    curve,   the 


RAILROAD   SURVEYING. 


137 


value  of  Ti  =  805.2,  and  for  a  4°  curve  7  =  201.3.  Now,  if  the 
distance  from  the  beginning  to  P.  I.  was  3346  ft.,  i.  e..  the  P.  I. 
was  at  station  33  +  46,  the  P.  C.  will  be  located  at  (3346  —  201.3) 


LU 


3144.7,  that  is,  at  station  31  +  44.7.  The  P.  T.  will  be  located 
an  equal  distance  from  the  P.  I.  or  at  3547.3,  according  to  the 
preliminary  survey.  Now,  length  of  curve  =  16  -f-  4  =  400  ft. 


138  SCRl'EYOR'S    HAND    BOOK. 

Then,  according  to  the  location  survey,  the  P.  T.  will  be  located 
at  3144.7  -f-  400  =  3544.7,  or  2.6  ft.  nearer  the  beginning  by  the 
curve  route  than  by  the  P.  I.  route.  Station  30  was  52.7  ft. 
from  this  P.  T.  according  to  the  preliminary  survey,  but  by 
the  location  chaining,  the  point,  instead  of  being  3,600  ft.  from 
the  beginning,  will  be  at  3597.4  ft.,  and  hence  the  station  stake 
36  will  be  taken  up  and  moved  forward  2.6  feejt,  so  that  it  will 
really  be  3600  ft.  from  the  beginning. 

PROBLEM  67. — The  P.  I.  in  the  preliminary  was  2614  ft.  and 
7  =  24°.  Find  the  positions  of  the  P.  C.  and  P.  T.  for  a  3° 
curve. 

PROBLEM  68. — The  second  P.  I.  in  the  previous  problem  was 
0(554  ft.  Find  the  position  of  P.  C.  in  the  location  survey  for 
a  3°  curve  if  7  =  18°. 

139.  Field  Book. — It  is  important  that  the  note  book  or 
field  book  should  be  neat  and  accurate  and  should  show 
all  the  necessary  data  for  the  location  of  a  curve  and  how  it  is 
connected  to  the  tangent  points,  where  it  begins  and  where  it 
ends.  The  supreme  test  of  note  taking  and  note  keeping  is  that 
ANY  engineer  can  understand  fully  and  accurately  exactly  what 
the  data  mean.  Fig.  74  is  an  illustration  of  both  pages  (left 
and  right)  of  a  location  survey  notebook  where  a  curve  has  been 
run  in  to  connect  two  intersecting  tangents.  The  angle  of  in- 
tersection of  the  tangents  7  —  12°  54',  and  the  tangents  are  united 
by  a  2°  30'.  The  length  of  tangent  for  a  1-degree  curve  for 
7  =  12°  54'  is  647.8  and  for  a  2°  30'  curve  the  length  of  tan- 
gent =  647.8 -=-2.5  =  259.1.  This  length  of  tangent  can  be  cal- 
culated from  the  following  formula  : 

50  tan  $  1     50  tan  6°  27' 
T=R  tan  \  i^-^^^-^^^-  =259'1' 

The  curve  is  to  begin  at  station  64  +  13.3  and  the  transit  is 
set  up  at  this  point  (the  P.  C.),  the  verniers  brought  to  zero, 
and  a  back  sight  taken  on  the  last  hub.  The  next  station  in 
advance  of  the  P.  C.  to  locate  is  65,  which  is  (6500  —  6413.3) 
86.7  ft.  from  the  P.  C.  For  a  full  100  ft.  the  deflection  is  half 
the  degree  of  curve  or  1°  15',  and  for  86.7  it  is  86.7 -r-  100  of 


RAILROAD   SURVEYING. 


139 


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140  SURVEYOR'S    HAND    BOOK. 

1°  15'.  Hence  the  deflection  =.867X75'  =  !°  05',  and  this 
should  be  recorded  opposite  the  station  65  that  it  locates.  For 
the  full  stations  66,  67,  68  and  69  the  record  in  the  "index" 
column  should  be  2°  20','  3°  35',  4°  50'  and  6°  05',  respectively, 
which  are  obtained  by  adding  1°  15'  to  the  record  of  the  last 
full  station  in  the  index  column.  Now,  the  length  of  the  curve 
=  100  X  12°  54'  -7-2°  30' =  516  ft.  Adding  this  516  to  the  64  + 
13.3  (the  station  number  of  the  P.  C),  we  get  69  +  29.3,  which 
is  the  station  number  for  the  P.  T.  The  deflection  angle  for 
the  29.3  =  . 293-X  75' =  22',  which  is  added  to  the  index  of  the 
last  full  station,  6°  05',  gives  an  index  of  6°  27'.  Now,  the  read- 
ing of  6°  27'  on  the  P.  T.  should  be  half  of  I,  that  is,  if  we 
double  the  index  for  the  P.  T.,  we  should  get  the  value  of  /. 
or  2(6°  27')  =  12°  54',  which  affords  an  easy  and  effective  check. 

It  may  happen  that  in  running  the  curve  the  transit  has  to 
be  moved  from  the  P.  C.  to  some  station  as  67,  the  index  of 
which  is  3°  35'.  Now,  after  setting  up  the  transit  over  67, 
we  can  back  sight  on  ANY  station,  provided  we  set  the  vernier 
to  read  the  index  of  the  station  sighted  at.  Thus,  if  we  backsight 
on  65,  with  telescope  reversed,  the  vernier  must  read  1°  05'  (on 
the  correct  side  of  the  vernier).  Then  to  locate  station  68,  all 
we  have  to  do  is  to  revolve  the  telescope  and  set  the  vernier 
at  4°  50',  the  index  of  the  station  sighted  at,  and  have  the  stake 
driven  at  this  point.  However,  if  we.  should  set  up  the  instru- 
ment at  67  and  backsight  on  64  +  13.3  (P.  C.)  with  telescope 
reversed,  we  must  set  the  vernier  at  0°  00',  the  index  of  the 
P.  C,  and  then  to  locate  68  we  again  make  the  vernier  read 
4°  50',  the  index  of  the  station  sighted  at.  Thus,  wherever  we 
set  up  the  transit  on  the  curve,  the  back  sight  on  any  station 
must  read  the  index  opposite  the  station  sighted  at,  and  to  lo- 
cate any  other  station  ahead,  revolve  the  telescope  and  set  the 
vernier  to  read  the  index  for  that  station. 

140.  Transit  Party. — The  transit  party  in  the  field  should 
consist  of  transitman,  rear  chainman,  head  chainman,  rear 
flagman,  stakeman.  and  axmen.  The  transitman  has  charge 
of  the  party  and  should  provide  himself  with  the  transit. 
tripod,  plumb-bob,  reading  glass,  notebook  and  pencil. 


RAILROAD   SURVEYING.  141 

The  rear  chainman  should  have  charge  of  chain  or 
tape  and  be  responsible  for  it.  The  head  chainman  should 
provide  and  take  care  of  the  flag  or  range  pole.  The 
stakeman  provides  bag  of  stakes,  keel  for  marking  same,  ax  or 
hatchet  for  driving  stakes,  and  tacks  for  hub-points.  The  rear 
ilagman  has  the  silent  duty  of  remaining  ever  in  readiness  to 
be  called  upon  to  give  a  sight  at  a  signal  or  call  from  the  transit 
man,  and  the  axmen  should  have  good  4.5-lb.  axes  to  clear  the 
way.  It  is  poor  economy  to  be  restricted  in  the  number  of 
men  that  are  to  do  the  clearing. 

141.  Stakes. — The    stakeman     should    provide    a    sufficient 
number    of   stakes    for   each    day's    supply   at   least.     The    stakes 

.      vary    in   size    (Fig.    75),    but    sawed 
~~M      stakes    are    2x1    ins.    by    18    ins.    in 
length,   while   "hubs"   should  be  2x2 
ins.   by   18  ins.   in  length.     The  flat 
shaped    stake    is    used     to     facilitate 
V      marking,  as  the  broad  surface  offers 
sufficient    space    for    the    number    of 
station  and  the  letter  indicating  the 

-,     line  to  be  written  or  printed  on  the 

1   i     stake.      The    figures    or    letters    are 

V'     printed  with  keel   (red  chalk),  which 

can  be  secured  from  dealers  in  draw- 
l^'       '  ing  supplies  or  from  local  hardware 

dealers. 

142.  Hubs. — At    every    angle    point    or    transit    station    a 
"hub"  is  located.     This  consists  of  a   stake   (Fig.  76),  2x2  ins., 
driven  flush  writh  the  surface  of  the  ground.     A  tack  is  driven 
in   the  top  of  the  hub,  where  the  range  pole  or  flag  rested  in 
the   line  of  sight.     After  the  tack  is   driven  partly  in  the  hub 
it   should   be  checked  by  the   transitman   so  that   any  error  in 
location  can  be  corrected  before  it  is  driven  too  far  to  be  with- 
drawn.    After  it  has  been  checked,  it  is  driven  flush  with  the 
surface  of  the  hub.     About  1  ft.  to  the  left  of  the  hub  a  "guard" 
stake   is   driven  with   the   number  of  the   station  marked  on  it. 
This  guard  stake  is  inclined  towards  the  hub  and  is  left  project- 


142  SURVEYOR'S    HAND    BOOK. 

ing  from  the  ground  several  inches,-  as  shown  in  Fig.  76.  The 
number  of  the  station  of  the  hub  should  be  marked  on  the  guard 
with  a  good  system  of  letters.  These  figures  should  be  printed 
with  red  keel,  and  in  no  case  should  they  be  written  with  a 
rough  figure  or  letter.  With  care  and  a  little  practice  the  stakes- 
man  can  soon  learn  how  to  mark  these  in  a  standard  and  sys- 
tematic way. 


Fig.    76. 

143.  Hand-Level. — This  instrument.  Fig.  77,  is  about  6 
ins.  long  and  has  a  level  tube  or  vial  on  top.  Across  one  half 
of  the  clear  glass  at  object  end  a  horizontal  line  "is  drawn.  The 
image  of  the  bubble  tube  can  be  seen  on  half  of  the  glass  at 
object  end  of  tube,  as  it  is  reflected  by  a  prism.  The  ends  of 
the  tube  are  closed  with  plane  glass  and  a  semi-cir.cular  convex 
lens  at  end  of  eye-piece  or  eye-tube  magnifies  level  bubble  and 


Fig.    77. 

the  cross  wire.  The  cross  wire  is  fastened  to  a  framework 
under  the  level  tube  and  adjusted  to  its  place  by  the  screw 
shown  on  end  of  level  case. 

To  use  the  level,  hold  it  with  the  hands  so  that  the  eye-end 
is  next  the  eye,  then  move  it  until  it  is  approximately  hori- 
zontal. The  image  of  the  bubble  can  then  be  seen  on  half  of 
the  object-end  glass.  When  the  bubble  appears  on  the  horizontal 


RAILROAD  SURVEYING. 


143 


mark  or  wire,  the  line  of  sight  is  horizontal.  To  use  the  hand- 
level  it  is  necessary  to  know  the  height  of  your  eye.  Sight 
through  the  hand-level  and  bring  the  bubble  on  the  horizontal 
wire  and  note  the  point  on  the  ground  indicated  by  the  line 
of  sight.  Unless  unusual  refinement  is  necessary  in  taking  to- 
pography,, the  hand-level  will  subserve  all  necessary  requirements 
and  it  is  an  economical,  efficient  and  expeditious  instrument  for 
this  purpose.  In  railroad  surveying  the  line  of  survey  affords 
a  base  line  from  which  all  transverse  measurements  can  be  made. 
"The  topographer  determines  the  height  of  his  eye  when  stand- 
ing in  his  usual  attitude  and  then 
taking  a  position  on  the  line  of  sur- 
vey. A  BCD  (Fig.  -78),  he  selects  a 
direction  at  right  angles  to  the  line 
of  survey.  Bringing  the  level  to  its 
horizontal  position  and  noting  where 
the  line  of  sight  strikes  the  earth  at 
point  1,  he  paces  the  distance  from 
line  to  point  (48  ft.,  say).  At  point 
1  he  notes  that  the  next  line  of  sight 
strikes  ground  at  point  3,  etc.  This 
process  is  continued  until  the  terri- 
tory 200  ft.  on  each  side  of  the  line 
covered.  If  the  height  of  the  eye 


Fig.    78. 
the      position 


of 


is  5.2  ft.,  then  each  point  of  inter- 
section of  horizontal  line  of  sight 
with  ground  is  5.2  ft.  higher  than 
the  observer.  The  elevation  of  the  observer's  position  is  'known, 
or  can  be  ascertained  from  the  levelman's  notes,  and  hence  the 
elevation  of  each  point  located  can  be  determined  by  adding  or 
subtracting  height  of  eye. 

On  the  lower  side  of  the  line  it  is  well  to  have  a  rodman 
provided  with  a  rod,  graduated  to  half-feet,  at  least  12  ft.  in 
length.  If  it  is  desired  to  have  all  contour  points,  the  uniform 
height  of  eye  above  or  below  the  adjacent  points  in  any  one 
normal  line,  the  topographer  can  have  his  rodman  walk  away 
from  the  base  line  in  a  normal  direction  till  the  rod  reads  double 


144 


SURVEYOR'S    HAND    BOOK. 


the  eye  height.  If  other  points  are  located,  the  rod  is  read  by 
the  hand  level  and  the  reading  recorded.  The  topographer  ad- 
vances to  the  rodman'si  position  and  sends  him  on  further  out 
to  locate  other  points.  If  no  rodman  is  used  the  topographer 
can  pace  the  distance  in  the  normal  direction  to  some  point 
which  he  guesses  is  about  the  eye-height  below  his  position.  If 
the  line  of  sight  from  his  point  strikes  below  the  surface  in  the 
normal  line,  he  must  go  toward  the  base  line  till  the  line  of 
sight  strikes  the  point  on  the  base  line.  The  distance  from  the 
base  line  is  found  by  subtracting  or  adding  the  distance  between 
the  final  location  and  the  assumed  point  to  the  distance  from 
base  line  to  assumed  point.  With  a  little  practice  a  topographer 
will  soon  be  able  to  select  a  point  within  a  foot  or  so  of  the 
correct  point. 

144.  Slope  Stakes  in  Excavation. — In  excavation  in  earth- 
work the  cross  section  is  defined  by  the  roadbed  AB,  Fig.  70,  and 
the  side  slopes  AE  and  BC.  The  amount  of  slope  of  BC  is 
determined  by  the  ratio  of  BG  to  CG,  and  is  designated  by  s.  •  '.$ 
=  BG  H-  CG  =  tan.BCG.  .'.  BG  =  s.CG  =  sfa,  where  h, 
=  height  of  point  C  above  roadbed  AB  =CG,  and  h«  =  EF  height 
of  E  above  roadbed  AB. 


Fig.    79. 
I==AB  width  of  roadbed. 


Now 


.'.Distance  out  of  stake  point  C  — half  width  of  roadbed  plus 
slope  times  height  of  point  above  roadbed. 

The  center  cut  DK  =  c  is  already  known  before  any  attempt 
is  made  to  set  the  slope  stakes.  The  level  is  set  up  in  some 


RAILROAD   SURVEYING.  145 

convenient  position  and  a  rod  reading  taken  on  the  station  at  D. 
Let  1234  represent  the  horizontal  line  of  sight, 

in  —  D\  =  rod  readjng  on  station  D, 
.:  H  =  DK  +  Dl  =  c  +  m. 

Now,  to  locate  the  slope  stake  at  C  at  a  horizontal  distance  d 
from  D,  try  some  point  as  P  and  find  the  rod  reading  P2  —  /'. 

Then  2Q  —  P2  =  H  —  r. 

Now,  H  —  r  is  the  "surface  height"  of  the  trial  point  above 
roadbed  AB. 

Calculated  distance  out  dc  =-b  +  s  (H  —  r}.     But 
Measured  distance  out  dm  =  b  +  s  QT. 

Hence,  we  see  that  when  the  trial  point  is  too  near  the 
center  the  measured  distance  out  is  less  than  the  calculated  dis- 
tance out.  Try  some  point  S. 

dm  =  measured  distance  out  =  b  +  s  UX. 
d0   —  calculated  distance  out  =  b  +  s  US. 
•  '  •  The  measured   distance  out  is  greater  than  the  calculated 
distance  out  when  the  trial  point  is  too  far  out,  and  vice  versa. 
Hence,   if 

dm  >  dc,  come  in ; 

dm  <  dc,  go  further  out. 

Rule:  If  the  measured  distance  out  to  the  trial  point  is 
greater  than  the  calculated  distance  out,  come  in,  and  vice  versa. 

Slope-Stakes  in  Level  Sections.— If  DT ',  the  surface  of  the 
ground,  is  horizontal,  then  DK  =  TQ.  In  this  case  the  point  T 
will  be  the  stake  point. 

Its  distance  out,  KQ  =  KB+  BQ  =  b  +  s  TQ  =  b  +  s  DK 
=  b  +sc. 

Thus,  in  level  sections  the  distance  out  is  found  by  multi- 
plying the  center  cut  by  the  slope  and  adding  the  half  width 
of  roadbed. 

EXAMPLE:  If  center  cut  — 14.6  and  slope  £  —  3:2  and 
width  of  roadbed  =  18  feet,  then 

3 
Distance  out  =  9  +  -    14.6  =  30.9. 


140  SURl'RyQR'S    HAND    BOOK. 

Field  Methods. — In  the  field,  if  the  ground  is  inclined,  the 
usual  practice  is  first  to  find  the  distance  out  on  the  assump- 
tion that  the  ground  is  level.  This  simply  serves  as  a  guide 
and  useful  help.  If  the  ground  slopes,  the  distance  out  on 
the  upper  side  of  the  center  of  roadway  is  always  greater 
than  the  distance  out  in  a  level  section,  if  the  ground  slopes 
uniformly.  While  on  the  lower  side  the  distance  out  of  the 
slope  stake  is  less  than  the  level  d.  o. 

3 
EXAMPLE.     Given.?-  =9--    26=18;c=14.    The  level  was  set 

up  r,nd  the  rod  reading  on  the  center  was  7.2.     For  a  level  sec- 

3 
tion,  the  distance  out  =  9  +  77  X  14  =  30.     On  the  upper  side 

the  trial  point  was  selected  at  32  ft.  from  the  center  where  the 
rodreading  was  5.4  ft. 

H  =  U  +7.2  =  21.2 
//_,-  =  21.2  —  5.4  —  15.8 

3 
Calculated  d.  o.=9  +  -^  x  15.8=32.7 

Now,  the  calculated  d.  o.  is  greater  than  the  true,  hence 
the  trial  point  is  too  near  center.  Try  a  point  34  ft.  out  when 
rod  reads  5.0. 

H  —  r  =  2l.2  —  5.0=16.2 

g 

Calculated  d.  o.=8  +  -^x  16.2=32.3  ft. 

The  calculated  d.  o.  is  less  than  true  d.  o. 
The  second  trial  point  is  too  far  out.     Try  point  33,  where 
rod  reads  5.2. 

H  —  r  =  21.2  —  5.2  =  16.0 

3 
Calculated  d.  o.=9  +  -^  x  16=33.0 

This    location   is    correct. 

On  the  lower  side  the  distance  out  must  be  less  than  30, 
the  d.  o.  for  a  level  section.  Try  a  point  29  ft.  out,  where  rod 
reads  r  =  SA.  •,•  •'' 

H  —  r  =  21.2  —  8.4  =  12.8 


RAILROAD   SURVEYING.  147 

Calculated  d.  o.  =  I)  +  V  >'   12.8  =  28.2 

Hence,  the  trial  point  was  too  far  out.  Try  a  point  at  28.4, 
where  rod  reading  =  8.3. 

H—r  =  21. 2  —  8.3—  12.9 

3 

Calculated  d.  o.  =*  9  +  j  X  12.9  =  28.35 

The'  location   is    sufficiently   accurate. 
PROBLEM  69.    Center  cut  =  16.6,  2b  =  18',  s  =  3  -^  2. 
Rod   reading   on   center  =6.2.     A   trial   point    was   taken    at 
35,  where  a  rod  reading  was  5.0.     Is  the  trial  point  too  far  out 
or  in  ?     Answer  : 

If  the  trial  point  was  at  39.0  and  the  rod  reading  was  4.9, 
is  it  too  far  out  or  in  ?  Answer : 

If  the  point  was  36  and  the  rod  reading  4.8,  how  is  it? 
Answer  : 

PROBLEM  70. — In  the  following  table:' 
c  =  center  cut,   ' 
•in  =  rod  reading  on  center, 
dm  ==  true  distance  out  of  trial  point, 

•i"=  rod   reading  on  trial  point, 
dc  =  calculated  distance  out. 
Find  the  results  as  to  accuracy  of  location  point* 

r  dc  Result 
5.2 
4.8 
5.0 
7.6 
8.0 
7.8 
6.4 
4.0 
4.2 
4.1 
3.6 

^  3.8 

145.  Slope  Stakes  in  Embankment. — In  embankments  the 
road  bed  AB  is  usually  for  single  track  roads  14  ft.  wide  and  the 
slope  varies  from  1  :1  to  2:1.  However,  on  levees  the  slope  is  as 
flat  as  5:1, 


Number 

c 

m 

s 

b 

dm 

A 

12.8 

6.6- 

3/2 

9 

30.2 

B 

12.8 

6.6 

3/2 

9 

32.0 

C   

...12.8 

6.6 

3/2 

9 

30.6 

n  

.  .  .  12.8 

6.6 

3/2 

9 

26.0 

E 

12.8 

6.6 

3/2 

9 

27.0 

17 

.  .  .  12.8 

6.6 

3/2 

9 

26.4 

G   ..... 

.  .  .  8.6 

5.4 

1/1 

9 

18.0 

II 

8.6 

5.4 

1/1 

9 

19.5 

I    

...  8.6 

5.4 

1/1 

9 

17.0 

.   8(1 

5.4 

1/1 

9 

18.9 

K 

11.4 

4.8 

2/1 

9 

35.6 

L   

.  ..11.4 

4.8 

2/1 

9 

33.8 

148 


SURVEYOR'S    HAND    BOOK. 


The  roadbed  is  AB,  Fig.  80,  and  the  side  slopes  BC  and  AE. 
Slope  stakes  must  be  set  at  the  foot  of  the  slopes  at  E  and  C. 
The  center  fill  KD  —  c  is  known  and  it  is  required  to  locate 
these  slope  stakes.  The  level  is  set  up.  1,  2,  4,  3  is  the  hori- 
zontal line  of  sight,  the  rod  reading  (m)  on  the  center  is  Dl. 
The  height  of  instrument  (H.  /.)  above  roadbed  AB  is  KI. 

Now,  BG  —  s  X  CG  and  AB  =  '2b,  DI=m 
H.I.  =  KI  =  m  —  c 

Distance   out  =  KG  =  KB  +  BG  =  b  +  s.  CG  —  b  +  sh 

The  rod  reading  (r)  on  C  is  4C, 

But  4C  =  4C  +  GC 

.'.,-  =  H.  I. +  h 

or  h  —  r  —  H  I'==r  —  m  +  c 

Distance  out  =  b  +  s  (r  —  m  +  r) 

Suppose  we  try  a  point  P  that  is  too  close  to  the  center. 

Rod   reading   (;-)=P2  =  r 


Fig.    80. 

dc  =  calculated  distance  out  =  b  +  s(r — m  +  c)  =  b+s.PQ. 
But  dm  =  true  distance  out  =  KQ  =  KB  +  BQ  =  b  +  sQT. 
Thus  dc  is  greater  than  dm. 

'Hence  the  calculated  distance  out  is  too  great  and  the  trial 
point  is  too  near  the  center. 

Try  a  point  5*  where  rod  reading  =  35". 

dc  =  b  +  s(r  —  m.  +  c)  =  b+s.US 

But  dm  =  KU  =  b  +  s.UX  ,£ 

. ' .  dc  is  less  than  dm. 


RAILROAD   SURVEYING.  149 

Hence  the  calculated  distance  out  is  less  tlian  the  true  dis- 
tance out,  or  the  trial  point  is  too  far  out.  Thus  we  see  that  the 
same  rule  applies  to  fills  that  applies  to  cuts. 

In  deep  fills  the  line  of  sight  1243  may  he  below  A 13  and 
the  height  of  instrument   (H.  /.)   will  be  negative.     In  this  case 
H.    I.  —  c  —  m 

Distance  out  =  b  +s  (c  —  m  +  r~) 

Example.— Given    2b=U:    j  =  3/2;    center    fill  =14.8     ft.; 
rod  reading  on  center  =  5.4.     If  the  ground  is  level  the  distance 
otit=7  +  3/2    (14.8)  =29.2  ft.     On  the  lower  side  the  distance 
out  will  be  greater  than  this,  while  it  will  be  less  on  upper  side. 
Try  a  point  31  out  where  the  rod  reading  =  7. 20. 
dc  =  7  +  |(14.8  —  5.4  +  7.2)  =  31.9. 
:.  Point  was  too  far  out. 
Try  a  point  32  ft.  out  where  r  =  7.3. 
d  =  7  +  3/2  ( 14.8  —  5.4  +  7.3)  =  32.05. 

The  location  is  sufficiently  accurate  for  practical  or  ordinary 
requirements. 

PROBLEM  71. — In  the  following  table  determine  the  results  of 
the  trials,  i.  e.,  whether  trial  point  is  too  far,  too  near,  or  cor- 
rect : 

dm  r          de  Results 

36  7.4 

27.2        7  5 
38  7.6 

22.0        5.6 
20.5        5.5 
20.7        5. 
23.0        3.4 

146.  Berms. — It  is  often  necessary  to  excavate  the  earth  near 
the  foot  of  the  slope  of  the  embankment  to  secure  enough  dirt  to 
make  the  embankment.  When  such  is  the  case  it  is  necessary  to 
leave  a  strip  of  unbroken  original  surface  at  least  4  ft.  in  width 
between  the  borrow  pit  and  the  foot  of  slope  to  afford  a  break 
for  earth  that  washes  down  or  off  the  slope.  Thus  in  Fig.  80 
FE  is  the  berm,  a  strip  of  undisturbed  natural  earth,  between 
the  embankment  CBAE  and  the  borrow  pit  NF. 

In  cuts  it  is  often  of  the  utmost  importance  to  have  an  un- 
disturbed natural  surface  on  each  side  of  the  cut.  To  do  this  it 


Number 

/ 

c 
...178 

m 
52 

s 

3/2 

b 

7 

B  .... 
C  .... 
D  .... 
E 

....17.8 
....17.8 
....14.4 
144 

5.2 

5:2 

4.8 
4.8 

3/2 
3/2 

1/1 
1/1 

7 
7 
7 

p 

14.4 

4.8 

1/1 

7 

C  .... 

....  9.2  , 

4.6 

2/1 

7 

150  SURVEYOR'S    HAXD    BOOK. 

is  necessary  to  prevent  the  deposition  of  any  excavated  mate- 
rial within  6  ft.  of  the  edge  of  the  side  slope.  If  the  loose 
earth  is  piled  near  the  edge  of  the  slope,  heavy  rains  will  wash 
it  down  the  slope  into  the  cut. 

Bibliography. — "Railroad  Location  Surveys  and  Esti- 
mates," by  F.  Lavis.  Published  by  the  Myron  C.  Clark  Publish- 
ing Co.  This  book  is  a  complete  epitome  of  actual  field  engineer- 
ing and  includes  a  history  of  the  preliminary  survey  from  the 
organization  of  the  party  to  the  completion  of  the  line.  No 
better  description  can  be  applied  to  this  work  than  to  say  that  its 
theme  is  to  tell  and  show  "how  to  do  things."  In  many  re- 
spects it  covers  a  territory  heretofore  not  traversed,  and  is  re- 
plete with  valuable  suggestions  gained  by  experience  as  a  field 
engineer. 

"Field  Manual  for  Railroad  Engineers.''  By  James  C.  Nagle. 
Published  by  John  Wiley  &  Sons,  403  pp.  One  of  the  leading 
field  books  of  the  country,  containing  full  directions,  suggestions, 
tables  for  the  solution  of  the  usual  problems  met  with  in  field 
operations  in  preliminary  and  location  surveys.  A  full  set  of 
tables  of  trigonometric  functions,  of  a  1°  curve,  transition  curve, 
coordinates,  squares  and  cube  roots. 

"Railroad  Curves  and  Earthwork."  By  C.  Frank  Allen.  Pub- 
lished by  Spon  &  Chamberlain.  490  pp.  Contains  discussion  of 
the  usual  railroad  curves  including  the  transition  curve,  rather 
full  treatment  of  slope  stakes  and  earthwork  problems,  with 
diagrams  to  facilitate  the  calculation  for  earth  work;  field  and 
office  tables. 

"The  Field  Engineer."  By  W.  F.  Shunk.  Published  by  1). 
Van  Nostrand  Company.  389  pp.  This  work  treats  of  the  prob- 
lems of  preliminary  and  location  surveys,  many  illustrative  ex- 
amples, the  essentials  of  slope  stake  setting,  and  the  usual  tables 
necessary  for  an  engineer  in  the  field. 

"Field  Engineering."  By  Wm.  H.  Searles.  This  has  been 
for  years  one  of  the  standard  manuals  for  field  and  office  engi- 
neers, and  it  covers  the  problems  of  railway  surveying,  location 
and  construction.  The  book  is  f.illy  illustrated  and  has  many 
valuable  tables  to  shorten  the  labor  of  calculation. 


CHAPTER   IX. 
EARTHWORK. 

147.  Prismoidal  Formula. — Let  Fig.  81  represent  ;•  solid 
bounded  by  two  parallel 
planes  and  whose  side 
faces  are  triangles.  Draw 
the  mid-section  12345678 
and  join  any  point  P  in 
this  mid-section  with 
ABCDEFGH,  1,  2,  3,  4, 
5,  6,  7,  and  8.  This  di- 
vides the  solid  into  three 
kinds  or  types  of  pyra- 
mids. The  first  class 
has  P  for  a  vertex  and 
A  BCD  for  a  base;  the 
second  has  P  for  a  ver- 
tex and  EFGH  for  a 
base,  while  the  third 
class  has  P  for  a  vertex 
and  for  bases  the  side 
face  triangles,  as  P  — 
EDC. 

Let  5,  =  area  A  BCD 
B,  =  area  EFGH 

h  =  perpendicular      distance     between     parallel     planes 
A  BCD  and  EFGH. 

1.  Volume  P—ABCD  =  J  ABCD  x  \h  =  \h  Bl 

2.  Volume  P-EFGH  =  J  EFGH  x  %h  =  \h  B, 

3.  To  find  the  volume  of  the  pyramids   of  the   third  class, 
consider  P  —  EDC  as  a  type  of  the  third  class.     The  pyramids 
P  —  £12  and  P  —  EDC  have  the  same  vertex   P  and   bases   in 
the  same  plane  EDC.     Hence  they  are  to   each  other  as   their 
bases, 

151 


152  SURVEYOR'S    HAND    BOOK. 


:.P  —  EDC  :  P  —  £12  :  :  EDC  :  £12. 

As  1  and  2  are  the  mid-points  of  the  sides  ED  and  EC,  EDC 
=  4X£12. 

:.P  —  EDC  =  \  X  P  —  £12. 

But  the  volume  of  the  Pyramid  P—  £12  =  $  x  Area  P12  X  \h 


/.  Volume  P—EDC=4x^X  P12=  -^  x  P12. 

Similarly,  Volume  P—EFC  =  -%  X  P23. 

6 

.-.  Total  volume  of  pyramids  of  third  class= 

=  -^-  (P12  +  P23-fP34  +  P45  +  P56  +  P67  +  P78  +  P18)  =  ~Q-  M, 

where  M  =area  of  mid-section  1234567S. 

Adding  the  volumes  of  the  three  types  we  get  for  total  vol- 

ume V  =  Volume   of  solid  =    ^  (B{  +  4A/  +  B2)  ...........  (33) 

148.     Railroad  Excavation.  —  In 

railroad  earthwork,  cross-sections  at 
right  angles  to  the  center  line  of 
track  are  taken  every  100  ft.  Slope 
stakes  are  set  and  data  obtained  for 
calculating  the  volume  to  be  exca- 
vated between  the  two  sections  100  ft. 
apart.  Such  n  solid  is  bounded  by  a 
plane  roadbed,  two  parallel  end  areas, 
whose  planes  are  perpendicular  to  the 

planes  of  the  side  slopes,  while  the  upper  surface  is  terminated  by 
planes  that  are  either  triangular  areas  or  that  can  be  divided  into  tri- 
angles by  drawing  the  diagonals  as  D'C.  The  prismoidal  form- 
ula applies  to  such  a  solid.  Fig.  82  represents  the  part  of  the 
excavation  on  one  side  of  the  central  plane  of  roadbed.  BKK'B' 
represents  half  of  the  roadbed  between  cross-sections  DKBC 
and  D'K'B'C'.  To  find  the  volume  of  the  excavation  by  the 
prismoidal  formula  given  above,  it  is  necessary  to  find  the  areas 
of  the  ends  or  bases  and  of  the  mid-section. 


EARTHWORK. 


153 


149.  Level  Sections.  —  Where  the  intersection  of  the 
cross-section  plane  with  the  surface  of  the  earth  is  horizontal, 
the  section  is  said  to  be  level,  or  a  one-level  section. 

In   Fig.   83   AB  —  26,  DK  —  c,   CG  —  EF.     Now,    BG  =  sCG 


Area  EABC  =  V2(EC  +  AB)DK 

=  i  (2b  +  2sc  +  26)  c, 


(34) 


Fig.    83. 


SA,    j  =  3/2,    find    area    of 


EXAMPLE:     Given    26  =  18', 
section. 

Area  =  2bc  +  sca  =  18  X  8,1+  3/2  X  (8.4)  2  =  257.04  sq.  ft. 

150.  Two  Level  Sections.  —  When  the  surface  of  the 
ground  slopes  uniformly  transverse  to  the  roadway,  two  points 
established  on  the  surface  will  be  sufficient  to  determine  the 
cross-section. 


Then  area  ABCE  —  ECGF  —  BCG— AEF 

=  %  (A,  +  hz)    (26  +  s!h.+  s!h)  —  Vish,  —  Vzsh* 

=  b    (7u  +  A2)  +  sfhh, (35) 

The   center  cut   is  used  only  in  locating  the  slope  stakes  at 
C  and  E,  but  is  not  used  in  the  calculation  of  the  area. 


154 


SURVEYOR'S    HAND    BOOK. 


EXAMPLE:     Given  25  =  18,  j  =  3/2,  /i,  =  8.4,  /z,  =  6.6. 

Area  of  section  =  9 (8.4  +  6.6)  +  3/2(8.4  X  6.6)  —135  +  83.16 
=  218.16  sq.  ft. 

151.  Three  Level  Sections. — By  far  the  most  common 
and  usual  section  is  the  one  where  the  two  side  heights  and  the 
center  cut  are  used  in  calculating  the  area. 

O 


J! 


Fig. 


As  usual,  CG  =  hlt  EF  =  h->, 
—  Ib,  BG  =  shl}  FA  =  sh,,  KG  = 
Area  DKBC  —  DKC  +  CKB 
In  the  same  way,  DKAE 
Total    area  =  r/2(d!  +  d,)  + 


=  d,,  KF  —  d,,  DR—c,  AB 
+  sh,f  KF    +  q  _  sh*. 
Vzcd,  +  %5/i,. 
,  +  Vzbh,. 

+  M  ....  ........     (36) 


Thus,  in  the  three-level  section,  the  double  area  is  equal  to 
the  center  cut  multiplied  by  the  sum  of  the  distances  out,  plus 
the  half  roadbed  multiplied  by  the  sum  of  the  side  heights. 


Fig. 

152.  Irregular  Sections. — When  the  surface  of  the  ground 
is  very  irregular,  rod  readings  must  be  taken  at  every  change 
in  slope  of  surface.  Thus,  in  Fig.  86  rod  readings  must  be 
taken  at  seven  different  places,  and  this  section  would  be  called 
a  seven-level  section.  In  the  field  we  would  locate  N  by  meas- 
uring its  distance  out  Kn,  and  by  its  elevation  Nn  above  AB 
the  roadbed.  Thus,  for  any  point  or  the  surface,  we  have  its 
co-ordinates,  i.  e.,  distance  above  AB  (roadbed)  and  the  dis- 


EARTHWORK.  155 

tancc  from  K  (center  of  roadbed)  to  foot  of  perpendicular.  To 
find  the  area  of  the  section,  we  find  first  the  area  on  the  right 
of  the  central  plane  DK  and  then  on  the  left. 

BKDMNPC  =  KDMm  +  mMNn  +  nNPp  +  pPCG  —  BCG 
Let  ct  hm,  hn,  V  h\   be  the  heights  of  D,  M,  N,  P,  C   above 
AB  and  dm,  dn,  dv  and  d\  equal  the  distance  out  of  M,  N,  etc. 
Area  KDMm=\  (c  +  hm)  dm 
Area  mMNn=b  (hm  +  &n)  (dn—dm) 
Area  nNP p=%  (hn  +  hp)  (dv—dn) 
Area  $PCG=\  (hp  +  hj  (</,— dp) 
Area    BCG  =^  (d{-  b) 
Expanding  and  simplifying,  we  have, 
Double  Area  BKDMNPC=cdm  +  hmdn  +  hndv  +  hv  dl  +  bhi— 

hidp—h9  dn—hn  dm (37) 

153.  Rules. — The  notes  in  the  field  book  are  written  as 
follows : 

Center  Side 

c_  km   hn   ^p   hi     o_ 

o  dm   dn   dp   d\     b 

Now,  the  point  B  is  a  corner  of  the  polygon  whose  area  we 
wish.  In  the  table  of  notes  we  write  each  cut  as  a  numerator 
of  a  fraction  with  the  distance  out  of  the  point  as  denominator. 
To  complete  the  notation  for  each  point  we  can  write  the  notes 
as  follows: 

c      hm    hn    hp    h\      o 
o'    dm'   dn'   dp    d\     b 

By  an  inspection  of  the  formula  for  the  area  in  connection 
with  the  Figure  86,  we  observe  that  each  positive  term  consists 
of  each  cut  or  height  (numerator)  multiplied  by  the  next  denom- 
inator to  the  right  (left),  and  that  each  negative  term  consists 
of  the  numerator  multiplied  by  the  denominator  to  the  left 
(right).  This  gives  the  following  usual 

Rule:     To  obtain  the  area  of  the  eross-section: 

1.  For  positive  terms,  begin  at  center  and  multiply  each  nu- 
merator by  the  next  oittin'ard  denominator. 

2.  For  negative  terms,  begin  at  ends  and  multiply  each   nu- 
merator bv  the  next  denominator  towards  center  cut. 


150  SURl'EYOR'S   HAND   BOOK. 

j.  Take  half  the  algebraic  sum  of  the  positive  and  negative 
terms  for  the  area  of  the  cross-section. 

The  data  should  be  arranged  as  in  Figure  87. 

If  we  begin  at  the  center  and  multiply  each  numerator  by 
the  denominator  with  which  it  is  connected  by  the  solid  arrow 
and  sum  the  results  we  get  the  positive  terms,  and  if  we  mul- 
tiply each  numerator  by  the  denominator  with  which  it  is  con- 


Fig.    87. 

nected  by  the  dotted  arrow,  we  get  the  negative  terms.  Half 
the  algebraic  sum  of  the  positive  and  negative  terms  gives  the 
area  of  the  cross-section.  Thus,  from  Fig.  87  : 

Double  area  on  right  =  14X  8  +  15  X  14  +  12  X  20  +  16  X  27 
+  12  X  9—12  X  20—16  X  14—8  X  12  =  542  sq.  ft. 

Double  area  on  left  =  14  X  7  +  0X21  +  8X0  —  8X7  =  303 
sq.  ft. 

Double  area  of  section  =542  +  303  =  845  sq.  ft. 

Second  Rule:  The  double  area  can  be  found  by  arranging 
the  data  as  in  Fig.  87  and  by  multiplying  the  sum  of  two  ad- 
jacent numerators  by  the  difference  of  their  denominators  and 
by  taking  the  algebraic  sum  of  the  products,  treating  the  two 
extremes  as  negative. 

Thus, 

Double  area  =  —  8  X  12  +  17  X  14  +  23  X  7  +  20  X  8  +27  X 
6+28  X  6  +  28  X  7—  12  X  18  =  845  sq.  ft. 

154.  Side  Hill  Cuts.  —  It  often  happens  that  the  railroad 
runs  along  the  side  of  a  hill  and  'that  part  of  the  roadbed  will 
be  in  cut  and  part  in  fill.  The  elevation  of  the  roadbed  is  known 
and  the  center  cut  or  center  fill,  as  the  case  may  be,  is  also 
known.  Thus,  if  EC,  Fig.  88,  is  the  surface  of  the  earth  and  AB 
the  roadbed,  part  of  the  cross-section  will  be  in  cut  and  part 
in  fill.  The  cut  DK  at  the  center  is  known  and  the  slope  stake 
at  C  is  located  as  usual.  The  point  P  (cross-section  grade- 


EARTHWORK. 


157 


point)  is  located  by  the  levelman  and  the  distance  KP  meas- 
ured. Below  the  point  P,  grade-point,  the  ground  shown  may  be 
roughened  or  cut  into  steps,  as  shown  in  figure,  to  prevent 
slipping  during  wet  weather. 


Fig. 
Let  BP  =  a,  then  area  PBCD  =  PDK  +  DKBC  =  4  (a— 6) 


Figr. 


Area  PAR  —  1/2EF  XAP  =  1/2  (2fc  —  a}  h,. 

EXAMPLE:  Given  ,2b  —  18,  j  =  3/2  on  both  sides  and  DK  = 
2',  h,  =  V,  //,,  =  —  4. 

The  distances  out  are  '21  on  upper  side,  and  !•">  on  the  lower. 
The  grade-point  is  found  3'  to  left  of  center. 

Area  in  cut  =  ^    (2  +  8)   +  *(2  +  3)  —  48  =  60  sq.  ft. 

Area  in  fill  =  1/2  6  X  4  —  12  sq.  ft. 

PROBLEM  72.— If  BK  =  S',  DK  =  2f,  KA~r,  PK  =  X, 
slope  in  cut  —  1  :1.  slope  in  fill  ==3:2,  find  area  in  cut  and  fill 
if  A1==8,  h,  =  —  4. 

155.  Average  End  Areas. — In  practice,  the  volume  is 
calculated  by  the  average  end  area  formula. 

Fig.  89  represents  a  form  of  a  three-level  section.  The  cen- 
tral plane  DK  divides  the  solid  of  excavation  into  two  parts  that 


158  SURVEYOR'S  HAND    BOOK. 

can  be  treated  separately.  Let  the  center  and  side  cuts  at  one 
station  be  c  and  hi  and  those  at  the  next  station  100'  away  be 
Ci  and  hi  and  let  both  sections  be  three-level  sections,  as  in  the 
figure. 

Let  Bt  =  area  DKBC 

B2  —  area  at  the  next  station  corresponding  to  DKBC, 

We  have, 


B,=  y2(d,'  d'  +  bh,') 

Now,  if  the  solid  is  bounded  by  plane  faces,  we  have  center 
cut,  side  height,  and  distance  out  at  mid-section. 


SM=(d,  +  <//)    (fl  +  r/)  -f  26  (A1  +  /it') 
But  V  =  true  volume.  =    -Q-  (B  +  4M  +  B)  ......  (33) 

=  -j2   (2c^i  +  2<f,'c1'  +  did'  +  d/c,  +  3W*i  -f  36/t/) 

The  average  end  areas  =  \  (Bl  +  #->)  ..................  (37) 

100  100 

Let  Ve  =  ~2~  (Bi  +  5,)  =  -^  (3diC1  +  3dl'clf  +  3bhl  +  3bhlf) 

156.  Error  of  Average-End  Area  Formula.  —  The  average 
end  area  formula  generally  gives  an  excess  of  volume.  Let  E 
be  the  excess  in  volume  by  end-area  formula. 

...  E=V&-V  =   -^-[(cj-c/)  (dl-dl')]  .........  (38) 

In  the  majority  of  cases,  d  —  c/,  and  di  —  rf/  have  the  same 
sign  -'-excess  is  positive,  that  is,  there  is  really  an  ex- 
cess. But  in  passing  over  a  saddle,  o  can  be  greater  than  c/ 
and  di  less  than  d\.  In  such  cases  the  excess  is  negative  —  that 
is,  the  volume  calculated  by  the  average-end-area  formula  is 
smaller  than  the  true  volume. 

By  common  consent  among  engineers,  contractors  and  sur- 
veyors, practically  all  volumes  in  railway  practice  are  calculated 
by  the  average-end-area  (AEA)  formula.  In  fact,  it  is  highly 
probable  that  for  the  real  earth  solid,  the  AEA  formula  gives 
results  as  near  the  actual  cubic  contents  as  the  true  prismoidal 
formula. 


EARTHWORK.  159 

157.  Examples. — The  stations  1,  2,  3,  etc.,  in  the  following 
table  are  100  ft.  apart.  The  numerators  in  each  case  show  the 
depths  of  cuts  and  the  denominators  the  distances  out  at  the 
different  points.  Width  of  roadbed  =  IS'.O,  slope  =  3/2. 

Cut  or  Fill  Cubic 

btation  Areas          Yards 

Left  c  Right 

1..  11  78       A2  221.31 

17.1  1003.3 

2..  _Li  QQ       —  320.46 

20.1  27.3  1381. 2 

3  JL-L   !5J?         9   ^       1?.8   HJ?       4>\5  4 

21.3    10.0  1LO    30.3  18197 

4.,    .  L2^  138       1*2   1^4 

27.3  12.0    33.6  233l  2 

5 m      16-6      is      7054 

In  calculating  the  areas  (as  at  Station  3)  we  arrange  the  data 
as  follows : 

'JL  _?J:  10.2  12.0  13^8  14.2  _0 

¥  2lT3  TOO  ~0~  TITO  3O3  "9" 

and  for  positive  terms  work  from  the  center  outward,  multiply- 
ing each  numerator  jy  the  next  denominator  ahead  as  we  pass 
out   from  center,  and   for   negative   terms    multiplying  each   nu- 
merator by  the  next  denominator  towards  the  center. 
Calculacion: 
Area  on  right=H12.0x  11.0+ 13.8  < 30.3+ 14.2x9.0— 14.2 x 

11.00H260.9 
Area  on  lef t  =  H12  x  10  +  10-2  X  21.3  +  8.2  x  9—8.2x10] 

=  164.53 

Check  calculation: 
Area  on  right  =  £  [  25.8  X   11.0  +  28  X  19.3  —  14.2  x  21,3] 

-=260.9  sq.  ft. 
Area  on  left—  £[22.2  x  10  +  '8A  x  H-3  —  8.2  X   12.3] 

=164.53  sq.  ft. 
.  Total  area  =  260. 9  +  164.53=425.4 


160  SURVEYOR'S  HAND   BOOK. 

Area  at  Sta.  4  =  £[13.8  X  12+15  X  33.6+  16.4  X  0  +  13.8  X 

27.3+12.!2  X  9—  12.0  x  16.4]  =  553.47. 
Area  at  Sta.  5  =  i[16.6  X  67.2  +  32.8  X  0]  =  705.4. 

100 
Volume     1-2    =  -^  [221.31  +  320.46]  =  1003.3  cubic  yds., 

Volume    2-3    =  -gy  [320.46  +  425.4  ]  =  1381.2  cubic  yds., 

1  no 
Volume    3-4    =  -^  [425.4     +  553.47]  =  1812.7  cubic  yds., 

Volume    4-5  .=  -^  [553.47  +  705.4  ]  =  2331.2  cubic  yds. 
Total  volume  1-5  =  6528.4  cubic  yards. 

PROBLEM  73. — Find  the  areas,  volumes  and  total  volume  from 
the  following  field  notes: 

Cut  or  Fill.  Cubic 

Station.  Areas          Yards 

Left  c  Right 

15.8  20.2 


32.7  18'4       39.3 

14.8  18.8 
3L2  16'9       37.2 

12.8    14.7  16.3    17.4 

28.2    13.0  15'°        12.0    35.1 

11.4  14.4    16.2 


J.O.O  11    /  \     o«~T 


.26.1  11.0    33.3 

Total  volume  =  7533.7  cubic  yards. 

158.  Preliminary  Estimates. — Tn  comparing  preliminary 
surveys  of  several  lines,  it  is  necessary  that  we  know  the  num- 
ber of  cubic  yards  of  excavation  required  on  each  line.  The 
preliminary  profile  will  give  the  cut  or  fill  at  the  different  sta- 
tions, and  if  we  assume  that  the  cross-section  is  level  we  can 
obtain  a  close  approximation  to  the  true  areas  and  hence  to 
the  volumes  without  going  to  extra  expense  of  setting  slope 
stakes  to  determine  the  true  cross-section. 

From  article  149  the  area  of  B  of  a  level  cross-section  is 
given  by 


EARTHWORK.  161 


Where  2b!=  width  of  roadbed,  r  =  center  cut,  s  Aslope 

If  2fe  =  18',  j  =  3:2,  then  #  =  18c  +  1.5r. 

Now,  if  we  make  c  =  l,  C,  3,  4,  5,  etc.,  we  get  areas  of  19.5,  .42, 
67.5,  96,  etc. 

It  is  assumed  that  any  of  these  areas  is  the  average  of  the 
two  sections,  50  ft.  on  each  side  of  it. 

R*it  Volume  =     ^    cubic  yards 

Making  B  equal  to  the  areas  above,  we  get  the  volumes  in 
cubic  yards  to  be  72,  156,  250,  356,  472  cubic  yards,  etc.  In  the 
same  way  we  can  find  the  volumes  for  any  width  of  roadbed 
and  any  slope.  The  usual  widths  are  12,  14,  16,  etc. 

Table  V.  gives  the  volumes  in  cubic  yards,  slopes  1  :4,  1  :2,  1  :1, 
3:2,  2:1  and  3:1  and  for  the  various  widths. 

EXAMPLE:  If  2b  —  l8,  ,y  =  l:l,  and  it  is  desired  to  find  the 
volume  in  cubic  yards  from  stations  5  to  stations  10,  where  the 
center  cuts  are  6;  8,  10,  12,  11,  we  look  in  the  table  headed 
"Slopes  3:2"  under  "base"  and  opposite  6  we  find  600,  oppo- 
site 8,  889,  etc.  These  are  read  from  Table  V.  and  recorded 
as  below: 

Station.  Center  Cut.  Volume. 

5  6  600 

6  8  889 

7  10  1,222 

8  12  1,600 

9  11  1,406 
10  9  1,050 

Sum   of   volumes  =  6,767   cu.   yds. 

From  this  we  must  subtract  half  the  end  volumes,  or  825. 
Volume  between  Sta.  5  and  Sta.  10  =  5942  cu.  yds. 

PROBLEM  74.—  If  the  center  cuts  at  Stations  17,  18,  19,  20 
and  21  are  12,  14,  15,  16,  15,  find  the  number  of  cubic  yards 
between  Stations  17  and  21  for  level  sections  by  use  of  Table  V. 

159.  Earthwork  Note-Book.  —  The  preliminary  estimate  of 
the  amount  of  earthwork  is  for  a  basis  of  comparison  with  other 
preliminary  lines,  but  the  final  estimate  is  based  on  the  actual 
notes  taken  in  vthe  field  in  setting  the  slope  stakes.  The  level 


162 


SURVEYOR'S  HAND   BOOK. 


notebook,  as  commonly  used,  has  a  left-hand  page  ruled  into  six 
columns,  as  shown  in  Fig.  90.     The  grade  column  (marked  "Gr." 


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Fig.    90. 

in  Fig.  90)    is  filled   in   from  the  profile  or  established  rise  per 
100   feet.    The   right-hand  page   of  the  notebook  is  ruled   intq 


EARTHWORK.  ,  163 

spaces  one-tenth  to  one-fourth  inch  square.  A  central  line  di- 
vides the  right-hand  page  into  two  halves  and  these  can 
he  utilized  for  the  earthwork  notes.  The  difference  between 
the  "Elev."  and  "Gr."  is  record  as  the  center  cut  under  "C." 
with  a  zero  for  the  denominator  and  the  left  and  right-side 
cuts  are  written  on  the  left  and  right,  respectively,  with  the 
"distance  out"  as  the  denominator.  The  areas  are  calculated 
in  square  feet  and  recorded  under  the  heading  "Areas"'  and  the 
volumes  are  found  by  the  mean-end-area  formula,  that  is,  by 
multiplying  the  average  of  the  two  end  areas  by  100  to  obtain 
the  cubic  feet,  and  by  dividing  this  by  27  to  obtain  th^  cubic 
yards.  In  passing  from  cut  to  fill  the  usual  practice  has  been 
followed,  averaging  the  plus  area  (cuts)  between  Stations  27 
and  28  to  obtain  the  amount  of  cut  or  plus  volume.  In  the  same 
way  the  average  of  the  negative  areas  between  Stations  2$  an^l 
29  has  been  taken  for  the  amount  of  fill  between  these  two 
stations.  Thus,  the, amount  of  cut  between  Stations  27  and  28  — 

100 

~2     (25.6  +  2.4)  -4-  27  =-•  51.9  cubic  yards      The  number  of  cubic 

1  00 

yards  of  fill  between  stations  28  and  29  =   -^  (7.9  +  12.8)  -s-  27 

=  38.3  cubic  yards. 

If  a  grade-point  occurs  between  two  stations  and  the  inter- 
section of  cut  and  fill  is  approximately  normal  to  line  of  sur- 
vey, that  part  in  cut  is  treated  as  a  wedge  whose  volume  is 
equal  to  the  half  area  in  cut  at  last  station  in  cut  multiplied 
by  the  distance  of  grade-point  from  said  station.  In  the  same 
way  the  part  in  fill  'is  treated  as  another  wedge  whose  volume  is 
found  the  same  way. 

160.  Special  Case. — Where  there  is  a  rather  sudden 
change  from  cnt  to  fill  a  special  solution  is  required  to  obtain 
the  exact  quantity  of  earth  in  cut  and  fill.  Let  AB,  Fig.  01.  be 
the  contour  between  the  excavation  •  and  embankment,  EB== 
width  ~f  roadbed  in  cut,  and  C//  =  width  of  roadbed  in  fill. 
Locate  the  points  A,  B,  C,  D  on  the  ground  where  the  plane 
of  the  roadbed  intersects  the  surface  of  the  ground.  Take  level 
notes  on  the  cross-section  FEE  and  CHG  and  measure  the  dis- 


164 


SURVEYOR'S   HAND   BOOK. 


54- 


tances  AE  and  DH.  Then  the  volumes  of  A  —  FEB  and  D  — 
CHG  are  treated  as  pyramids  whose  bases  are  FEB  and  CHG 
and  altitudes  AE  and  DH,  respectively.  The  volume  between 
the  sections  FMB  and  the  next  full  station  is  found  by  multi- 
plying the  average  of  the  end  areas  by  the  dis- 
tance between  FMB  and  the  full  station. and 
by  dividing  the  product  by  27.  Thus,  if  the 
contour  AB  occurs  between  Stations  54  and 
55  and  M  is  42  ft.  from  Station  54  and  the 
area  in  cut  at  Station  54  is  286.8  sq.  ft.,  area 
FMB  =  73.8 ;  then  the  volume  between  sta.  54 
and  FMB  =  Vz  (73.8  +  286.8)  X  42  -f-  27  = 
280.5  c.  y. 

EXAMPLE:      Given     £5  =  18',      C7/  =  14', 
slope  in  cut=l:l,  slope  in  fill  =  3.2,  cut  at  F 
=  6.00',  cut  at  Jl/  =  3.2,  fill  at  G  =  4.8',  fill  at 
K  =  2.2',  /*E  =  20.0',  Z?H  =  15.6.     Then  area 
of  FMB  =  73.8  sq.ft.    Volume  A  — FEB=%X  73.8X20.  ^27=18.2 
cu.  yds.     Area  CKHG~ 49.50  sq.  ft.    Volume  D  —  CKHG—Vz 
X  49.50  X  15.6  4-  27  =  9.53  cu.  yds. 

Now  distance  £.4  =  20  and  D//=15.6,  hence  MAr  =  10.0, 
A' A7"  =  7.88.  Therefore,  distance  from  K  to  station  54  =  42  +  10 
+  7.8  =  59.8.  Hence,  distance  from  K  to  station  55  =  100  —  59.8 
=  40.2  ft.  Find  number  cubic  yards  in  fill  between  K  and  55 
if  area  in  fill  at  55  =  222.2  sq.  ft. 

161.  Borrow  Pits. — When  the  excavations  will  not  fill  the 
cuts  or  embankments,  or  when  the  haul  is  too  far  for  economy, 
it  becomes  necessary  to  obtain  earth  from  the  areas  adjacent 
to  or  near  the  embankment.  Such  places  are  called  "borrow 
pits,"  and  when  it  is  desired  to  ascertain  the  amount  of  earth 
excavated  the  area  is  first  divided  into  rectangular  sections  about 
10x10  ft.  With  some  local  point  as  bench  mark  or  datum,  the 
elevation  of  each  corner  of  rectangles  is  determined  with  refer- 
ence to  the  bench  mark.  After  the  excavating  is  finished  the 
points  are  re-located  in  the  pit  and  the  new  elevation  of  each 
point  again  determined  with  reference  to  the  datum.  The  dif- 
ference of  the  two  elevations  of  any  point  will  be  the  depth  of 


EARTHWORK. 


165 


excavation  of  that  point.  The  volume  taken  out  of  any  rec- 
tangle will  be  found  by  drawing  the  diagonal  (as  13)  in  the 
1234. 


._£ 

/ 

a 

\ 

\-        & 

4 

3 

_& 

K 

4 

z.. 

Fig.    92. 
Then  let  A  =  area  1234, 

711  =  depth  of  cut  at  1, 

712  =  depth  of  cut  at  2, 
7/3  =  depth  of  cut  at  3, 
7?4  ~  depth  of  cut  at  4. 


Now 


A 
volume  1  2  3  =  ~7r 


volum 


134=    -~- 


.  '  .Total  volume  1234=   g-(/t2  +  ht  +  2^  +  2/*»)  ....  (39) 

Rule:  Multiply  one-sixth  the  area  of  rectangle  by  twice  the 
sum  of  the  two  heights  at  ends  of  diagonal  plus  the  sum  of  the 
other  tivo  heights.  Ordinarily,  the  volume  can  be  found  with  suf- 
ficient exactness  by  taking  the  average  of  the  four  cuts  and  mul- 
tiplying this  by  the  area,  or, 

A 

Volume  =  -^-  (hi  +  h.2  +  h3  +  7*4)  ,  nearly 

In  order  to  re-establish  the  points  1,  2,  3,  etc.,  after  the  ex- 
cavation has  been  made  it  is  necessary  to  establish  some  base 
line  like  PQ,  that  will  not  be  disturbed  by  the  plows  or  teams 
and  tie  every  point  to  this  line  by  rectangular  co-ordinates. 
Thus,  to  re-establish  the  point  3,  its  perpendicular  distance  from 
a  certain  point  on  PQ  must  be  known. 


IOC 


SURrEYOR'S   HAND   BOOK. 


162.  End  of  Fill. — When  a  fill  has  a  gap  in  u  for  a 
trestle,  the  dirt,  is  often  allowed  to  spill  obliquely  be- 
\ond  the  end  of  the  dirt  on  the  track  grade.  Thus,  if  A 13', 
Fig.  93,  is  width  of  roadbed  and  is  the  termination  of  dirt  road- 
bed, the  dirt  is  allowed  to  fall  down  the  slope  to  the  irregular 
line  CDEFG.  where  GH  and  CK  are  the  intersection  of  the 
side  slopes  with  the  ground  surface.  The  depths  of  G  and  C 
below  the  plane  of  ro?dbed  are  known  from  the  slope  stake 
notes.  It  is  sufficient  to  treat  the  volume  whose  base  is  ABDF 
as  a  wedge  whose  base  is  AMB  and  whose  edge  is  PEP,  and 
the  volumes  CBD  and  AFG  as  quarter  cones  whose  liases  are 
BCD  and  GAP,  and  whose  altitudes  are  the  heights  of  B  and  A, 
respectively. 

The  bases  BCD  and  AFG  can  be  treated  as  quarter-ellipses. 


Fig.    93.  Fig.    94. 

Hence,  the  area  BCD  =  &  BD  X  BC*  and  area  AFG  =  V±  AG 
XAF*. 

EXAMPLE:  Given  AB=-U'-  heights  of  A,  M,  and  #  =  8', 
7.6'  and  0.8',  respectively,  and  AC  =  12',  BC=W.2,  BD  =  U', 
AF—U'. 

Cross-section  of  wedge  =  14  (8  +  2  X  7.6  +  6.8)  =  10"). 

14 


Volume  of  wedge  =  — —  X  ^7  =  27.2  c.  y. 

Volume  of  CBD  =  J  X  l^Lll?  x  6J  _  9.4  c.  y. 

12  X   147T         8 
Volume  of   .4F£  = 


4 

12  X   14r         8 
X  £ X  ^  =  13.0  c.  y. 

Total   volume   GCDF  =  49.6   cu.   yds. 

163.  Overhaul. — In  contracts  for  earth  work  the  price 
per  cubic  yard  is  based  upon  the  condition  that  for  this  price 
no  material  should  be  transported  further  than  a  certain  dis- 


EARTHWORK.  16? 

tancc  (called  the  "free  haul"),  and  that  extra  pay  should  be 
allowed  for  all  material  carried  further  than  this.  "  Tn  Fig.  04 
.1JL  =  free  haul,  ABMN  represents  the  excavated  material  and 
LDllK  represents  the  material  deposited  in  embankment.  If 
O  and  P  represent  the  centers  of  gravity  of  the  volumes  ABMN 
and  LDHK,  the  distance  RQ  is  the  total  haul  and  the  excess 
of  this  over  the  free  haul  is  the  overhaul. 

.*.  Overhaul  =  RM  +  L Q 

To  find  the  centers  of  gravity  O  and  P,  multiply  each  ele- 
mentary mass  by  its  distance  from  some  point  C  and  divide  - 
the  sum  of  such  products  by  the  sum  of  the  elementary  masses.- 
However,  it  is  sufficient  in  practice  to  find  a  point  that  divides 
each  mass  into  two  equal  parts  and  use  these  as  t£e  centers 
of  gravity. 

164,  Shrinkage. — From  a  varied  mass  of  data,  H.  P.  Gil- 
lette, in  his  book  on  "Earthwork  and  Its  Cost,"  has  compiled 
the  deduced  principles  : 

1.  Taking   extreme   cases,   earth    swells    when   first    loosened 
with    a   shovel,   so  that  after  loosening  it  occupies   1  1-7   to   l1/^ 
times  as  much  space  as  it  did  before  loosening.     In  other  words, 
loose  earth  is   14  to  50  per  cent  more  bulky  than  natural  bank 
earth. 

2.  As   an  average,,  we  may  say  that  clean  sand   and  gravel 
swell  1-7,  or  14  to  15%;  loam,  loamy  sand  or  gravel  swell  1-5, 
or  20%;   dense  clay  and  dense  mixtures  of  gravel   and  clay,   % 
to  V-i,  or  33  to  50%,  ordinarily  about  35%  ;  while  unusually  dense 
gravel  and  clay  banks   swell  50%. 

3.  That    this    Irose    earth    is    compacted    by    several    means: 
(a)  the  puddling  action  of  water,  (b)  the  pounding  of  hoofs  and 
wheels,   (c)   the  jarring  and  compressive  action  of  artificial  roll- 
ing- 

4.  If  the  puddling  action  of  rains  is  the  only  factor,  a  loose 
mass   of   earth    will    shrink   slowly  back  to   its   original   volume, 
but  an   embankment   of  loose   earth   will,  at  the  end  of  a  year, 
be  still  about    1-12,  or  8%,  greater  than  the  cut  it  came  from. 

5.  If  the   embankment   is   made   with   small  one-horse   carts, 
or  wheel   scrapers,  at  the   end  of  the  work  it  will  occupy  5  to 


108  SURVEYOR'S  HAND   BOOK, 

10%  less  .space  than  the  cut  from  which  the  earth  was  taken, 
and  in  subsequent  >ears  will  shrink  about  2r/f  more,  often  less 
than  *l</c. 

(5.  If  the  embankment  is  made  with  wagons  or  dump  carts, 
and  made  rapidly  in  dry  weather  without  water,  it  will  shrink 
about  3%  to  !<>%  in  the  year  following  the  completion  of  the 
work,  and  very  little  in  subsequent  years. 

~.  The  height  of  the  embankment  appears  to  have  little  ef- 
fect on  its  subsequent  shrinkage. 

8.  By   the  proper   mixing   of  clay  or   loam   and   gravel,   fol- 
lowed  by  sprinkling  and   rolling  in  thin  layers,   a   bank  can  be 
made   weighing   1%    times    as   much   as   loose    earth,   or   133  Ibs. 
per  cu.  ft. 

9.  The   bottom   lands   of  certain   river  valleys   and  banks   of 
cemented  gravel  or  hardpan  are  more  than  ordinarily  dense  and 
will  occupy  more  space  in  the  fill  than  in  the  cut  unless  rolled. 

Earthwork  is  paid  for  by  the  cubic  yard,  usually  measured 
"in  place,"  that  is,  in  the  natural  bank,  cut,  or  pit  before  loosen- 
ing; but  there  is  no  good  reason  why  it  should  not  be  measured 
in  the  fill  or  embankment,  and  it  often  is  so  measured  where 
it  is  very  difficult  to  measure  the  borrow  pits.  In  either  case 
the  specifications  should  distinctly  state  how  the  'measurements 
are  to  be  made.  Sand  or  gravel  for  mortar  and  concrete  are 
usually  paid  for  by  the  load  in  the  wragon. 

Bibliography. — "Railway  and  Earthwork  Tables."  By  C. 
L.  Crandall.  It  is  sufficient  to  say  that  this  book  bears  out  its 
title,  where  the  tables  are  arranged  by  which  we  can  read  the 
volume  for  railroad  cuts  and  fills  for  any  of  the  usual  dnta 
given  in  the  field  notebooks  for  cross-sectioning. 

"Railway  Earthwork."  Parts  I  and  IT.  By  the  late  A.  M. 
Wellington.  Part  I  discusses  the  volumes  of  the  various  solids 
in  railway  earthwork,  while  Part  II  consists  of  a  series  of  dia- 
grams from  which  the  volume  corresponding  to  the  field  notes 
can  be  read  at  once. 

"Railroad  Curves  and  Earthwork"  (with  Tables).  By  C. 
Frank  Allen.  In  the  section  on  earthwork  the  theorv  and  use  of 


EARTHWORK.  10!) 

graphical  diagrams  are  treated  and  the  methods  of  using  these 
diagrams  to  obtain  the  volumes  are  illustrated  by  many  ex- 
amples. 

"Primoidal  Formulas  and  Earthwork."  By  T.  U.  Taylor. 
The  history  of  the  different  formulas  that  apply  to  the  earth- 
work solid  and  their  application  to  railway  cuts  and  fills  are 
given.  A  chapter  is  devoted  to  the  two-term  formula  wherein 
it  is  shown  that  there  is  an  indefinite  number  of  two-term  form- 
ulas that  give  the  exact  volume  of  the  prismoid ;  that  if  we 
take  the  average  of  two  sections,  these  sections  must  be  21.14 
feet  from  each  end  of  the  solid  100  ft.  in  length. 

"Manual  of  Road  Making."  By  W.  M.  Gillespie.  Contained 
in  appendix  some  40  pages  upon  the  subject  of  earthwork,  in 
which,  in  addition  to  the  treatment  of  the  ordinary  cases,  he 
showed  that  the  prismoidal  formulas  applied  to  give  the  exact 
volume  of  the  earthwork  solid  when  the  upper  surface  was  a 
warped  surface. 

"Earthwork  and  Its  Cost."  By  H.  P.  Gillette.  244  pages. 
This  work  has  taken  up  and  considered  actual  examples,  giving 
date,  size  of  contract,  conditions  under  which  constructed,  kind 
of  earth,  how  handled,  etc.  The  author  has  winnowed  from 
many  a  contract  the  -essentials  as  to  shrinkage,  classification, 
loosening,  cost  when  carried  by  wheel  barrows,  wagons,  buck 
and  drag  scrapers,  wheel  scrapers,  by  elevating  grader,  steam 
shovels,  cars,  etc. 

"Rock  Excavation.  Methods  and  Cost.''  By  H.  P.  Gillette. 
375  pages.  Its  title  abundantly  indicates  its  scope.  Its  estimates 
of  cost  are  from  concrete  examples  where  actual  conditions  are 
given. 

"Handbook  of  Cost  Data."  By  H.  P.  Gillette.  One  of  the 
most  valuable  books  for  the  engineer  that  has  appeared  in  many 
years,  and  it  comes  nearer  filling  a  long  existing  void  than  any 
book  before  the  engineering  public.  It  includes  a  great  deal 
of  the  material  in  the  two  books  mentioned  above  and  much 
additional  matter.  It  deals  directly  from  the  ground  with  such 
questions  as  co^t  of  earth  and  rock  excavation,  roads  and  pave- 


170  SURVEYORS  HAND   BOOK. 

ments,  stone  masonry,  concrete  construction,  water  works,  sew- 
ers, piling,  trestling,  erecting  buildings,  steam  and  electric  rail- 
ways, bridge  erection,  railway  and  topographic  surveys  and  mis- 
cellaneous structures.  This  book  should  be  a  valuable  Vadc 
Me  cum  for  any  engineer  who  has  to  deal  with  the  cost  of  struc- 
tures. 


CHAPTER  X. 
CITY    SURVEYING. 

165.  The  City  Engineer. — The  most  important  factor  and 
vital  unit  in  all  city  surveying  is  the  city  engineer.     A  careless 
engineer  means  a  careless,  loose,  inaccurate,  conflicting  and  liti- 
gous    survey.     The  city   engineer   is   the   supreme   court   and  all 
the   lower   courts   with   respect  to   the   accuracy  of   city   survey- 
ing.    As    the    city    engineer,    so    is    the    survey.     The     engineer 
should   be   the   first   instrument   of   precision   selected,   and   it   is 
supreme   folly   to   have   a   standardized   steel   tape   and   a   highly 
sensitive  transit  in  the  hands  of  a  careless  operator.     We  apply 
corrections  for  sag,  temperature  and  pull  to  our  tape-line  meas- 
urements, but  these  are  mockeries  if  the  engineer  can  be  sagged 
from  1ii's  true  course,  or  if  he  allows  a  "pull"  to  draw  him  from 
the   straight  line.     The  accurate,  just,  and  fearless  performance 
of  his  duty  should  be  his  pjatform.     To  this  end   should  he  be 
born,  for  this  cause  came  he  into  the  world,  and  he  should  bear 
witness  to  the  truth. 

The  surveying  dernanded.of  a  city  engineer  does  not  involve 
any  principles,  operations,  or  intricacies  that  may  not  be  easily 
overcome  by  any  person  who  understands  thoroughly  the  use  of 
the  ordinary  instruments  and  theory  of  surveying  heretofore  de- 
scribed, but  as  land  is  much  more  valuable  in  cities  than  in  the 
country  it  follows  that  the  measurement  of  city  property  must 
be  made  much  more  carefully  than  the  survey  of  a  farm.  The 
accuracy  of  the  survey  should  increase  with  the  value  of  the 
property.  Small  errors  that  may  be  neglected  now  may  involve 
perplexing  difficulties  in  years  to  come.  It  is  always  wise  and 
safe  to  be  considered  a  little  too  fine-haired  rather  than  a  little 
too  careless. 

166.  Objects  of  Survey. — The  prime  object  in  a  city  sur- 
vey  is   to   establish   the  points   and   boundaries   of  city  property 
with  absolute  accuracy.     To  do  this,  it  is  necessary  to  establish 
certain  reference  lines  or  points  which  will  remain  permanently 

171 


172  SURVEYOR'S  HAND   BOOK. 

fixed  and  which,  like  a  reference  library,  are  of  easy  access  and 
of  undisputed  authenticity.  Property  is  valuable,  and  to  prevent 
litigation  it  is  necessary  to  have  all  property  lines  authoritative- 
ly established  beyond  the  shadow  of  a  doubt.  Chains  with  their 
many  hundred  wearing  surfaces  are  unfit  for  such  work,  and  as 
it  lacks  accuracy  the  compass  can  not  be  used.  As  the 
ordinary  transit  measures  to  the  nearest  minute  and  as  an  angle 
of  1'  is  subtended  by  an  arc  of  18  ins.  at  a  distance  of  one  mile 
its  use"1  should  be  precluded  where  accurate  work  is  demanded. 
The  primary  object  of  a  city  survey  should  be  the  accurate  loca- 
tion of  all  property  lines  in  accordance  with  recorded  notes  or 
maps,  and  complete  provision  for  the  rapid,  convenient  and  ac- 
curate re-establishment  of  these  at  any  time.  The  most  accurate 
instruments  and  greatest  care  should  be  used. 

167.  Monuments. — It  is  of  fundamental  importance  that 
lasting  monuments  be  established  to  which  all  city  lines,  points 
and  buildings  can  be  referred.  Eternal  monuments  is  the  price 
of  accurate  work  in  city  surveying.  While  engineers  and  sur- 
veyors are  liable  to  rail  at  and  descant  sneeringly  at  the  loose 
methods  pursued  in  making  the  original  land  surveys,  many  of 
such  land  surveys  are  monuments  of  accuracy  when  compared 
with  the  surveys  of  many  of  our  cities.  In  fact,  although  our 
original  land  surveys  were  loosely  made,  all  transfers  of  property- 
have  been  based  on  such  surveys.  These  surveys  have  many 
monuments  in  the  shape  of  trees  to  stand  as  silent  witnesses  to 
be  called  upon.  The  land  at  least  had  an  original  survey,  while 
the  original  part  of  a  majority  of  our  cities  has  expanded  with- 
out the  semblance  of  an  original  survey.  It  is  worthy  of  re- 
mark that  more  care  and  accuracy  are  displayed  in  surveying 
the  "additions"  and  "out-lots"  than  obtained  in  the  original  sur- 
vey of  the  nest-egg  of  the  town. 

But  whether  or  not  monuments  were  established  in  the  origi- 
nal survey  of  the  town,  it  is  of  the  utmost  and  urgent  importance 
that  they  be  established  at  the  earliest  possible  moment.  In 
some  cities  a  very  loose  habit  has  prevailed  of  using  old  build- 
ings for  reference  points.  Such  a  practice  should  be  condemned 
as  a  make-shift,  for  with  the  enhanced  value  of  property,  such 


CITY    SURVEYING.  173 

buildings  arc  liable  to  be  razed  to  make  room  for  mocUrn  struc- 
tures. 

168.  Additions. — The    map    of    every    "addition"    or   pro- 
jected town  should  when  filed  in  the  county  clerk's  office  show 
clearly  the  location  of  all  monuments  and  no  map  should  be  ad- 
mitted to  record  that  does  not  give  these  data.     Not  only  should 
such  a  map  show  the  location  of  such  monuments,  but  a  full  de- 
scription of  such  monuments  should  be  made  a  matter  of  record. 
Such  requirements  should  not  be  a  matter  of  custom,  ethics,  or 
taste  of  the  surveyor,  but  should  be  a  matter  of  law;  and  there 
is    no    more   reason    for   a    law    authorizing   the    employment    of 
skilled  surveyors  to  locate  state  lands  and  file  a  complete  set  of 
field  notes  for  the  same  than  there  is  for  a  similar  law  requiring 
every  city  to  have  a  similar  map  or  set  of  notes  filed  and  made 
a  matter  of  record.     These  notes  should  be  so  clear  and  include 
such  a  number  of  sketches  that  they  may  be  readily  understood 
by  any  person  of  average  intelligence ;  and  such  notes  should  be 
capable  of  only  one  interpretation.     Litigation  has  always  fed  fat 
on  loose  and  inaccurate  surveying  and  an  unmonumented  city. 

Monuments  should  be  set  and  established  by  the  original  sur- 
veyor. He  it  is  that  made  the  surveys  with  respect  to  such 
monuments  and  it  is  his  duty  to  finish  his  survey.  Tt  can  be 
truthfully  said,  "An  unmonumented  city  has  no  survey."  There 
is  a  certain  respect  paid  to  the  County  Surveyor  and  his  work 
should  command  respect.  So  it  should  be  with  the  work  of  the 
city  engineer,  but  while  our  laws  provide  for  "witness  trees," 
"fore  and  aft  trees,"  for  land  surveying,  there  are  in  many 
states  no  adequate  laws  for  enforcing  or  establishing  imperish- 
able witnesses  to  the  city  lines  in  a  city  survey. 

169.  Kinds   of  Monuments. — Monuments    should  be  con- 
structed of  permanent  material  and  the  special  kind  will  be  de- 
cided  by   the   question   of   economy.     The   materials   most   com- 
monly used  are   stone,  concrete,  wood,   and  iron  rods  or  pipes. 
If  a  stone  is  used  it  should  be  imbedded  in  the  ground  with  its 
upper  part  well  underneath  the  surface,  so  that  the  big  end  will 
be  down  and  so  that  it  will  rest  solidly  in  its  bed  and  have  no 
tendency  to  change  its  position.     A  small  hole  from  %  to  1  in. 


174  SURVEYOR'S   JIAXD    BOOK. 

in  diameter  should  be  drilled  in  the  upper  surface  of  the  stone 
TO  a  depth  of  G  to  8  ins.  Into  this  hole  a  copper  bolt  should  he 
inserted  and  melted  lead  or  babbit  metal  run  around  it  to  hold  it 
securely  in  position.  The  upper  end  of  the  bolt  should  be  flush 
with  the  surface  and  two  normal  diametral  lines  should  be 
marked  across  the  bolt,  their  intersection  forming  the  reference 
point  over  which  the  plumb-bob  of  the  transit  is  suspended,  or  a 
{'UK  poie  set  when  other  points  are  to  be  established. 

A  concrete  block,  Fig.  0-">,  can  1>e  constructed  as  a  monument 
rind  it  has  many  advantages  over  the  stone  monument,  as  it  can 
he  formed  into  any  desired  shape.  For  economy,  the  concrete 
monument  should  he  built  in  the  form  of  the  frustrum  of  a 
cone  or  pyramid,  ami  its  upper  surface  should  be  kept  well  K 
low  the  surface  of  the  street.  The  copper  bolt  can  be  imbedded 

in  the  concrete  before  it  hardens 
and  it  can  be  located  in  any  de- 
sired  position  in  the  concrete. 

If  wood  is  used,  the  most  dur- 
able available  wood  should  be  se- 
lected.   The  important  monument 
should  be  at  least  6x6  ins.  i>y  J  ft 

in  length  and  should  be  imbedded 
Fig.    9;>.  6 

on    hard    soil    or    preferably    en 

a     flat     rock     or     a     concrete    mixture.      Cedar    is     an     excel 
lent  material,   while  osage   orange    (boisdarc)    has   no   superior. 
The  young  mountain  locust,  10  ins.  in  diameter,  is  the  most  dura- 
ble in   the   east,    while   mesquite   would   be   practically   the   only 
locally  available  wood  of  the  southwest. 

An-  iron  rod  or  pipe  is  often  driven  with  maul  or  sledge  for 
a  monument,  but  these  do  not  make  very  satisfactory  monu- 
ments and  are  not  to  be  recommended,  but  it  must  be  said  that 
they  are  infinitely  better  than  none  at  all  and  greatly  superior  to 
a  small  wooden  stake.  Wooden  stakes  are  very  easily  disturbed 
or  destroyed  and  unless  they  are  immediately  replaced  by  other 
monuments  of  a  more  permanent  character  the  work  will  be 
wasted. 


CITY    SURVEYING.  175 

If  the  street  is  already  graded  and  paved  the  monument 
should  be  set  with  its  top  below  the  foundation  of  the  pavement 
and  should  be  protected  and  made  easily  accessible  by  means  of 
an  iron  jacket  and  cover  plate  such  as  are  provided  for  the 
valves  of  the  city  water  supply. 

If  the  street  is  neither  graded  nor  paved,  some  thought 
should  be  given  to  the  probable  final  street  level  and  the  monu- 
ment should  be  located  to  cor.form  therewith  if  possible. 

It  is  the  duty  of  the  city  engineer  to  establish  suitable  per- 
manent   monuments    wherever    needed,    to    indicate    the    same 
clearly  and  correctly  on  the  proper  maps,  to  deposit  in  the  office 
a    complete    set    of    all    field 
notes,    to    leave    his    work    in 
such  a  state  that-  it  may  all  be 
intelligible    and    useful    to    his 
successor. 

170.  Location    of    Monu- 
ments.— These  should,  if  pos- 
sible, be  located   in  the  center 
lines     of     cross     streets     and 
should  be  on  high  points.  They 
should  be  of  easy  access ;  a  few 
well   located     monuments     are 
more    valuable    than    many    to 
which    ready    reference    cannot 
be  made.  The  fundamental  re- 
quisites of  good  monuments  are  that  their  location  is  known  and 
that  their  distance  and  azimuth  are  matters  of  record. 

Sometimes  it  is  impracticable  to  set  monuments  in  the  center 
of  the  street.  When  this  is  the  case,  they  should  be  placed  as 
near  the  center  as  convenient,  but  they  should  always  be  refer- 
enced in  to  the  four  corners  of  the  street. 

Wherever  the  monuments  are  located,  the  four  corners  of 
the  streets  should  be  marked  by  sub-monuments  whose  distances 
from  the  main  monument  are  recorded. 

171.  Tapes. — It  is  useless  to  have  an  excellent  system  of 
monuments  unless  this  excellence  prevails  throughout  the  whole 


17G  SURVEYOR'S   HAND    BOOK. 

organization  of  the  city  survey.  All  lines  should  be  measured 
with  standardized  steel  tapes.  The  material  of  the  tape  should 
be  of  the  best  steel  and  its  own  individual  constants  should  be 
determined.  It  should  be  sent  to  the  U.  S.  Coast  and  Geodetic 
Survey,  Washington,  D.  C,  to  be  standardized.  It  is  there  com- 
pared .with  an  absolute  standard,  its  coefficient  of  expansion 
ascertained,  its  pull  and  temperature  for  standard  length  de- 
termined. These  data  are  returned  with  the  tape  and  in  all  im- 
portant measurements  should  be  used  and  corrections  should  be 
made  for  temperature,  pull,  sag,  and  grade.  But  accurate  work 
can  not  be  performed  with  accurate  instruments  unless  accurate 
methods  are  used.  In  chaining,  if  the  street  is  graded  uniformly 
and  the  tape  can  be  made  straight,  the  correction  for  sag  would 
thus  be  eliminated.  If  in  addition  to  this,  the  standardizing  pull 
be  applied,  the  only  correction  remaining  would  be  that  for 
temperature  and  grade,  and  if  the  street  is  horizontal,  the  only 
correction  to  be  applied  would  be  that  due  to  temperature. 

172.  Transit. — After  the  monuments  have  been  located 
with  accuracy  and  the  exact  point  of  these  monuments  marked 
by  the  intersection  of  lines  on  the  copper  bolt  head,  it  becomes 
necessary  to  use  the  most  accurate  and  refined  instruments  in  the 
prosecution  of  the  further  surveying  of  the  city.  As  the  ordi- 
nary transit  reading  to  one  minute  of  arc  would  produce  an  error 
of  18  ins.  in  one  mile,  its  unfitness  for  accurate  city  surveying  is 
at  once  seen.  It  is  useless  to  locate  monuments  accurately  and 
to  use  an  accurate  standardized  tape  in  connection  with  a  transit 
that  has  such  possibilities  of  error  as  the  ordinary  engineer's 
transit.  For  this  reason  a  special  transit  (Fig.  97)  is  constructed 
with  minuter  graduations.  The  same  reason  that  precludes  the  use 
of  the  engineer's  transit  in  refined  city  work,  of  course,  would  ex- 
clude the  surveyor's  compass  to  a  greater  degree.  In  the  mod- 
ern transit  constructed  for  accurate  city  surveying,  the  needle 
and  the  needle  box  are  omitted  and  the  standards  are  construct- 
ed in  one  U-shaped  piece  that  gives  greater  rigidity  of  bearing  to 
the  horizontal  axis  that  supports  the  telescope,  and  consequently 
greater  accuracy.  The  horizontal  circle  is  much  larger  and  the 
graduations  can  be  made  as  small  as  ten  seconds  of  arc.  The 


CITY.  SURVEYING. 


177 


Fig.   97. 


178 


SURVEYOR'S  HAND   BOOK. 


horizontal  circle  is  protected  from  view  by  a  cover  plate  except 
where  the  slot  is  made  for  the  reading  by  the  verniers.  The 
verniers  are  read  by  special  reading  glasses,  which  are  often  at- 
tached to  the  instrument  itself.  Whatever  the  fineness  of  the 
reading  may  be,  whether  it  reads  to  10"  or  2<>",  we  can  by  the 
repeating  method  read  the  angle  five  times  and  thus  reduce  the 
fineness  of  the  reading  to  one-fifth  of  that  given  by  the  verniers. 
Thus  if  the  transit  is  graduated  to  30",  we  can  by  repeating  the 
observation  five  times  get  a  reading  of  6",  and  if  it  reads  to 
1<>",  we  can  by  the  repetition  of  five  times  get  a  reading  to  2". 
In  the  length  of  one  mile  a  reading  of  2"  would  mean  about  a 
half  an  inch  error. 

The  transit  can  be  provided  with  stadia  wires  and  complete 
vertical  circle  and  a  heavy  tripod. 
The  complete  vertical  circle  and 
stadia  wires  are  auxiliaries  that  are 
added  for  the  purpose  of  making 
topographic  survey.  The  transits  ful- 
filling these  requirements  cost  from 
$300  to  $700  and  if  it  is  desired  the 
stadia  wires  and  vertical  circle  can 
be  omitted. 

173.  Datum.— There  should  be 
established  in  every  city  bench 
marks  to  which  all  elevations  should  be  referred.  In  the 
majority  of  cases,  the  elevation  of  the  bench  marks  can  be  re- 
ferred to  the  sea  level  or  mean  low  tide.  In  many  cities  the  U. 
S.  Coast  and  Geodetic  Survey  has  bench  marks  with  reference  to 
sea  level  that  have  been  established  by  a  system  of  precise  levels 
run  and  checked  from  the  coast  to  the  interior.  These  are  by 
far  the  most  reliable  and  accurate  bench  marks  that  can  be  ob- 
tained. The  U.  S.  Geological  Survey  has  also  a  chain  of  bench 
marks  established  in  certain  sections  of  the  country.  The  bench 
marks  established  by  these  two  surveys  are  often  copper  bolts 
set  vertically  in  the  cap  stone  of  bridge  piers,  or  horizontal  bolts 
set  inside  of  stone  buildings.  Another  form  is  a  circular  disc, 
Fig.  98,  from  the  center  of  which  a  bolt  3  ins.  long  projects  at 


Fig. 


CITY    SURVEYING.  179 

right  angles  to  the  surface  of  the  disc.  Two  diametrical  lines 
normal  to  each  other  are  marked  across  the  face  of  the  disc  and 
the  elevation,  is  stamped  on  the  horizontal  line  of  the  disc.  A 
bed  or  setting  is  cut  out  of  the  stone  for  the  disc  and  in  the 
center  of  this  bed  a  hole  is  drilled  to  receive  the  bolt.  The  bolt 
is  then  leaded  into  the  stone. 

174.  General  Maps. — There  is  generally  a  small  scale  map 
made  of  the  whole  city,  but  this  shows  few  engineering  features 
and  except  in  the  case  of  small  cities  it  can  not  show  the  dimen- 
sions of  lots  and  the  field  notes  for  the  location  of  monuments. 
In  addition   to    the   map    of   the    whole   city   there    should   be   a 
map  of  certain  sections  to  a  scale  sufficiently  large  to  show  all 
lengths  of  all  lines  and  angles  made  by  intersecting  lines.     It  is 
the  practice  in  many  cities  to  have  block  maps  containing  from 
one  to  four  blocks  with  the  position  of  all  monuments  marked 
with  distance  from  street  corners  and  angles  made  by  such  tie 
lines.     These  maps  should  show  the  center  line  of  street,  angles 
of  intersection  of  center  lines,  and  the  location  of  monuments  on 
street  corners. 

The  map  should  contain  the  following  data : 

1.  Length  of  all  lines. 

2.  Angles  made  by  intersecting  lines. 

3.  The  exact  position  of  all  monuments. 

4.  The  number  of  each  block  and  lot. 

5.  The  names  of  all  streets  and  streams. 
f>.  Water  pipes  and  fire  plugs. 

7.  Sewer  pipes. 

8.  The  true  meridian. 

9.  Width  of  streets. 

10.  The  position  of  adjoining  property  lines. 

11.  A  complete  title  to  map. 

12.  The  scale. 

175.  Water-Pipe  Map. — If  the  city  owns  the  water-works 
and  sewerage  systems,  it  should  possess  an  up-to-date,  accurate 
and  distinct  map  of  both  the  water-pipe  lines  and  the  sewer-pipe 
lines.     If  the  city  is  small  and  pipe  connections  are  not  intricate 


180  SURVEYOR'S  HAND  BOOK. 

nor  numerous,  one  map  will  suffice  for  both  systems,  by  adopting 
a  different  legend  for  the  two  systems.  A  water-pipe  map 
should  show  clearly  the  position  of  all  mains,  valves,  connections, 
fire  hydrants,  size  of  pipe,  and  all  side  connections.  Such  a  map 
usually  pays  for  itself  many  times  over  and  it  is  a  very  loose 
city  government  that  does  not  keep  such  a  map.  Without  a 
pipe-line  map  all  extensions  and  repairs  have  to  be  made  some- 
what upon  the  temporary  makeshift  basis.  In  some  cases,  the 
city  authorities  depend  upon  the  memory  of  a  day  laborer  to  lo- 
cate sub-mains,  and  these  often  have  to  spend  hours  in  search 
of  the  pipe,  all  of  which  time  could  be  saved  by  an  accurate  map. 
If  a  private  company  owns  the  water-works,  an  accurate  map  is 
part  of  its  equipment  because  it  is  simply  a  part  of  good  busim  -< 
to  have  such  a  map.  However,  there  often  seems  to  be  some- 
fatality  about  municipal  ownership  in  regard  to  proper  records 
The  city  records,  covering  expenditures  of  millions  of  dollars  for 
public  improvements  are  often  thrown  aside  or  dumped  into 
boxes,  or  cases  that  cannot  be  used  for  any  other  purpose.  The 
proper  keeping  of  engineering  data  is  a  weak  spot  of  municipal 
ownership,  an  indictment  that  cannot  obtain  in  the  same  degree 
against  private  ownership. 

When  city  streets  are  improved  by  paving,  it  is  of  the  utmost 
importance  in  making  repairs  or  connections  to  know  the  exact 
distance  of  the  main  or  sub-main  from  the  sidewalk  or  property 
lines,  as  it  is  a  matter  of  economy  in  time  and  renders  the  tear- 
ing up  of  a  large  area  of  paving  unnecessary. 

176.  City    Blocks. — The    size    and    shape    of   city    blocks 
vary  in  different  sections  of  the  country  and,  in  fact,  in  different 
sections  of  the  same  city.     It  is  difficult  to  set  any  limits,  but  the 
regular  rectangular  blocks  vary   in    length   from  400  to  900   ft. 
With  a  width  of  street  of  80  ft.  there  will  be  5%  to  11  blocks  to 
the  mile,  and  of  course  if  the  streets  are  narrower  there  would 
be  from  6  to  12  to  the  mile,  etc. 

177.  Rectangular  Blocks.— In   ordinary   cases,   a   rectangular 
block  consists  of  two  rectangular  sections  with  an  alley  between. 
Thus  if  ABFG,  Fig.  99,  is  a  rectangular  block,  there  are  two  sec- 
tions, ABCD  and  EFGH,  with  an  alley  DCEH.    If  the  length  of 


CITY    SURVEYING. 


181 


the  block  is  300  ft.  and  if  each  section  contains  five  lots,  these 
should  be  60  ft.  wide.  The  length  of  the  lot  is  125  ft.  and 
width  of  alley  16  ft.,  the  block  being  266  ft.  wide. 

Each  lot  is  described  (1)  by  its  number,  (2)  by  the  number 
of  the  block,  (3)  by  the  sub-division  or  addition,  (4)  by  the 
name  of  the  city,  county,  and  state.  Thus  we  should  write : 

•  "Lot  number  (3)  three  in  Block  thirty-nine  (39),  Borden 
Addition,  in  City  of  Austin,  Travis  County,  Texas."  This  de- 
scription is  sufficient  if  an  official  map  of  this  "Borden  Addition" 
is  on  record  in  the  city  or  county  clerk's  office,  showing  all  di- 
mensions of  such  lot.  However,  if  it  is  desired  to  insert  the 
metes  and  bounds,  this  can  be  done  as  follows : 

"Lot  number  three  (3)  in  block  thirty-nine  (39),  Borden 
Addition,  in  the  City  of  Austin, 
County  of  Travis,  State  of  Texas, 
and  bounded  as  follows :  Beginning 
at  the  northeast  corner  of  lot  num- 
ber two  (2)  in  said  block,  addition 
and  city,  one  hundred  and  twenty 
(120)  feet  from  the  northwest  cor- 
ner of  said  block,  thence  S  9°  W, 
one  hundred  and  twenty-five  (125) 
feet  with  the  east  line  of  lot  num-  Ross  ST: 

ber    two    (2),    to    a    corner    on    the 

alley,  thence  S  81°  E  sixty  (60)  feet  to  the  SW  corner  of 
lot  number  four  (4)  ;  thence  N  9°  E,  with  west  line  of  lot  number 
four  (4)  one  hundred  and  twenty-five  (125)  feet  to  a 
point  on  the  north  side  of  block,  the  northwest  corner  of  lot 
number  four  (4),  thence  N  81°  W  with  the  north  line  of  said 
block  and  with  the  south  line  of  Adams  St.,  sixty  (60)  feet  to  the 
beginning." 

178.  Rectangular  Lots. — The  size  of  lots  runs  the  scale 
from  the  narrow  business  property  lot  25  ft  in  width  to  that  of 
the  broad  frontage,  merging  into  the  suburban  property  defined 
by  the  acre  and  metes  and  bounds.  The  lots  in  the  regular  resi- 
dence section  vary  from  40  to  100  ft.  in  frontage,  but  there  is 


18.! 


SURyEYOR'S   11AXD    BOOK. 


infinite  variety  to  the  special  dimensions  and  the  foregoing  fig- 
ure are  approximate  only. 

In  regard  to  the  depth  of  the  regular  rectangular  residence 
lot,  it  can  be  said  that  the  depths  are  approximately  double  the 
frontage,  varying  from  90  to  200  ft.  unless  some  irregular 
boundary,  stream  or  hill  intervenes  to  modify  the  general  plan 
by  which  the  lots  are  laid  off. 

179.  Irregular  Blocks  and  Lots. — Tt  often  happens  that 
the  topography,  old  roads  or  streams  force  the  engineer  to  make 
a  block  of  irregular  shape,  the  flat-iron,  horse-shoe,  triangular  or 

oval.  In  such  a  case  no 
rules  can  be  laid  down  for 
cutting  such  a~  block  up  into 
lots,  and  the  engineer  can 
have  only  one  guide,  and 
that  is  to  make  each  lot 
wide  enough  for  the  build- 
ings of  that  locality  (busi- 
ness or  residence)  and  of 
the  ordinary  depth. 

If  ABCD,  Fig.  100,  rep- 
resents the  apex  block  be- 
tween two  converging 
streets  it  is  often  difficult  to 

OT:  divide   this   up   into  lots  to 

the    best     advantage.      The 

simplest  method  is  to  run  the  lot  side  lines  perpendicular  to  the 
street  line.  This  is  shown  by  the  side  lines  of  lots 
1,  2,  3,  4,  and  5,  all  of  which  lines  are  perpendicular 
to  the  street  line  on  Shaw  St.  However,  it  may  hap- 
pen that  for  some  substantial  reason  the  lot  lines  are  parallel  to 
the  alley  or  some  other  line.  Again  the  lines  may  be  drawn  ac- 
cording to  no  system  whatever.  In  the  latter  case,  the  opposite 
sides  of  the  lot  will  not  be  parallel,  and  it  will  be  necessary  to 
describe  each  lot  by  the  metes  and  bounds.  In  addition  to  this 
the  corners  should  be  marked  by  some  permanent  marks,  as  gal- 
vanized pipe,  stones  or  concrete  blocks. 


CITY    SURI'EYIXG.  183 

In  the •  flatiron  form  of  blocks,  as  in  Fig.  100,  a  dead-end 
alley  can  be  provided  for  at  the  big  end  of  the  block,  and  this 
can  extend  as  far  as  the  line  of  lots  will  permit.  A  lot  in  an 
irregular  shaped  block  should  have  a  rather  full  description. 
Thus  lot  9  should  be  described  as  follows  :  "Lot  number  nine 
(9)  in  Block  thirty-five  (35),  Division  A,  in  the  City  of  Austin, 
County  of  Travis,  State  of  Texas,  which  is  bounded  as  follows : 
beginning  at  an  iron  pipe  in  line  of  Fox  Street  70  ft.  from  north- 
west corner  of  said  block  35,  thence  along  Fox  Street  S  6°  W  40 
ft.  to  corner  of  lot  number  8,  thence  S  87°  E  64  ft.  to  a  copper 
bolt  in  a  stone  which  is  a  corner  to  lots  number  2  and  9  of  said 
block,  thence  north  46  ft',  to  a  stone  corner  to  lots  3,  9,  and  10, 
thence  S  87°  15'  W  54  ft.  to  the  beginning." 

180.  Private  Notes.— The  careful  engineer  will  mark  the 
length  of  all  lines,  the  angles  made  by  the  boundary  lines  of  lots, 
give  the  full  number  of  lot,  the  name  of  "addition,"  and  all 
other  data  necessary  to  define  clearly  and  distinctly  the  lot  so 
that  another  engineer,  years  later,  will  have  no  trouble  in  tracing 
the  steps  of  the  former.  Every  modern  engineer  experiences  a 
genuine  appreciation  of  the  original  engineer,  when  he  finds  that 
the  recorded  map  shows  clearly  all  distances  and  angles,  and  the 
modern  does  not  hesitate  to  commend  the  former  when  map 
dimensions,  when  applied  to  the  field,  are  found  to  be  true.  Too 
many  engineers  are  stingy  with  their  data  when  it  comes  to  put- 
ting it  on  the  map.  The  question  often  arises  as  to  how  much 
data  should  be  placed  on  the  map,  and  this  can  be  answered  by 
saying  that  sufficient  data  should-  be  placed  on  the  map  to  enable 
another  engineer  to  go  upon  the  ground  and  re-locate  any  lot 
without  doubt  or  shadow  of  turning.  Until  this  condition  is 
fulfilled  the  map  is  incomplete;  the  claim  of  the  engineer  that 
his  notes  are  private  cannot  be  set  up  or  maintained.  The  city 
engineer  "is  a  public  officer  and  should  keep  complete  records  af 
all  work  done  in  his  official  capacity  during  his  incumbency.  If 
he  walks  out  of  his  office  and  retains  notes  the  lack  of  which 
would  embarrass  his  successor,  he  is  practical^"  a  thief."  (Ernest 
McCullough.) 


184  SURVEYOR'S  HAND   BOOK. 

181.  Prescriptive  Rights. — Owners  in  new  and  sparsely 
settled  additions  arc  often  permitted  to  locate  their  own  lots, 
and  in  doing  so  they  get  the  side  lines  of  the  lots  shifted  a  few 
feet.  A  fence  is  usually  erected  on  the  lot  lines  erroneously  lo- 
cated and  this  fence  stands  as  the  visible  mark  of  the  lot  lines 
for  many  years.  The  adjacent  lots  are  not  improved  and  the 
result  is  that  the  owner  of  the  improved  lot,  although  his  fence 
lines  are  wrongly  located  and  though  there  may  be  an  excess  in 
his  frontage,  has  been  in  peaceable  possession,  undisturbed  for  a 
sufficient  time  to  constitute  a  prescriptive  right.  This  gives  him 
the  right  of  possession  and  when  the  owners  of  adjacent  lots 
want  the  amount  their  deeds  call  for,  they  find  the  prescriptive 
right  set  up  as  a  bar  to  moving  fence  lines.  The  result  is  that 
legal  mills  have  to  be  set  to  grinding  with  no  assurance  of  the 
quality  of  the  grist. 


Fig.   101. 

Where  the  prescriptive  right  obtains  it  is  of  the  highest  im- 
portance to  property  owners  to  see  that  their  lots  are  located 
properly  and  accurately  by  an  official  engineer,  and  that  perma- 
nent corners  are  established. 

182.  Cross-Section  of  Streets. — After  the  blocks  and  lots 
have  been  laid  off  and  accurately  marked,  it  then  often  falls 
within  the  province  of  the  surveyor  or  engineer  to  establish  the 
form  of  cross-section  of  the  street.  This  cross-section  is  usually 
a  curve  having  a  certain  rise  or  crown,  depending  on  the  mate- 
rial out  of  which  the  surface  of  the  street  is  constructed.  If  the 
street  is  paved  with  vitrified  brick  the  crown  should  be  from  % 
to  %  in.  per  foot  of  half  width.  Thus  for  a  street  width  of  96 
ft.  between  side  walks  there  should  be  a  crown  of  6  to  18  ins., 
preferably  the  latter.  If  the  side  walks  are  at  different  eleva- 
tions, local  conditions  may  demand  that  the  cross-section  shall 
consist  of  two  curves  tangent  to  each  other  at  the  crown  or 
crest  and  that  the  amount  of  their  descent  shall  be  different. 


CITY    SCRl'EYIXG.  185 

Thus  in  Fig.  101  the  cross-section  can  be  formed  by  the  two 
curves  OA  with  a  fall  of  Ol~  and  OB  with  a  fall  of  OC. 

Let  VA  =  distance  from  curb  to  crest  =  b  ;  OV"==v,  OK 
=  .r;  PK  =  y,  fall  from  O  to  P. 

Then  if  curve  OA  is  a  parabola. 


Or  y  =  £F*2  .................................  (40) 

If  b  =  48  feet,  and  v  =  1  8"  ^.  1  .5',  y  =  "4^^  **  -  J^T)' 

3t^ 

If  y  equal  fall  in  inches  and  x  =  distance  in  feet,  j  =  y-jg 

By  making  x  =  0,  4,  8,  etc.,  the  falls  at  these  distances  are 
found  below. 

.r  v  x  y  x  y 

0  :<><>  16  2.00  36  10.12.") 

4  .12--,  24  4.5  40  12,5 

8  .5  32  8.00  48  18.0 

Formula  (40)  is  a  general  formula  and  will  apply  to  any  con- 
ditions, and  does  not  assume  that  the  crest  O  is  in  the  center  of 
street. 

Circular  Curve.  —  Some  engineers  prefer  to  treat  the 
curve  OA  as  a  circle  and  specify  the  amount  of  curvature  by  the 
radius  of  the  circle. 

Let  r.4=half  of  chord  of  circular  arc  OA  ;  r  =  rise  =  OF. 
As  the  arc  is  very  flat,  KP  can  be  treated  as  a  secant  from  P  to 
circle. 

Then  if  R  =  radius  of  circle, 
OK^  —  KP  (2R  +  PK),  or, 
x*  —  y  (-2R  +  3.)  =  2Ry  +  y": 

The  last  term  is  so  small  in  comparison  with  the  first  that  it 
can  be  omitted. 


If  the  crown  is  %  or  Vs  in.  per  horizontal  foot,  then  j?  =  192 
ft.,  or  576  fl.  respectively. 

183.  City  Engineering  Records.  —  There  are  three  differ- 
ent kinds  of  records  that  should  -be  kept  by  the  City  Engineer: 


18<>  SURlrEYOR'S  HAND   BOOK. 

I     Field  Note-Books. 

II     Detail  maps. 

Ill  Orders,  letters  of  correspondence,  bids,  prices,  contracts, 
specifications,  results  of  tests,  etc. 

184.  Field  Note-Books. — For  simplicity  one  kind  of  style 
of  book  that  is  applicable  to  all  kinds  of  surveying  should 
be  adopted  and  used  exclusively.  It  should  have  stiff  covers, 
should  be  leather  bound,  and  be  as  large  as  the  average  coat 
pocket  will  accommodate.  If  the  left  hand  page  is  ruled  with 
"horizontal  blue  lines  V±  in.  apart  and  the  page  divided  by  vertical 
red  lines  into  six  columns,  the  right  page  being  divided  into 
small  squares  by  horizontal  and  vertical  blue  lines,  with  a  verti- 
cal red  line  in  the  center  of  the  page,  the  book  will  be  found  to 
answer  admirably  for  all-round  work.  In  this  book,  level  notes, 
transit  notes,  notes  on  earth-work,  sewer-pipe,  water-pipe,  tri- 
angulation,  land  surveying,  etc.,  can  be  recorded  with  clearness 
and  neatness.  The  measurements  can  all  be  placed  on  the  K'ft 
page,  while  sketches  can  be  placed  on  the  right  page  to  an  ap- 
proximate scale. 

Proper  provision  should  be  made  for  storing  or  filing  all  the 
note-books,  preferably  in  a  fire-proof  vault.  The  books  should  be 
numbered  consecutively  and  arranged  in  order  on  the  shelves, 
and  the  Chief  Engineer  should  require  every  note-book  to  be 
put  in  its  proper  place  on  the  shelves  or  in  the  vault  over  night. 
Books  should  be  assigned  to  certain  classes  of  work  rather  than 
to  particular  assistants  or  transit  men.  -  For  example,  all  mis- 
cellaneous work  relating  to.  property  lines  should  be  kept  in  one 
book,  all  work  relating  to  grades  of  streets  in  another,  etc. 
Each  new  book  should  be  immediately  given  a  number,  the  class 
of  work  for  which  it  is  intended  being  plainly  lettered  on  the 
outside  of  the  cover,  thus :  "Street  Grades  and  Profiles,  5th,  Oth 
and  7th  Wards,"  and  the  first  half  dozen  pages  should  be  left 
blank  for  an  index  to  its  contents.  Every  new  piece  of  work 
should  be  indexed  in  the  book,  and  also  in  the  general  index  of 
all  the  note-books  kept  in  the  office.  The  Chief  Engineer  should 
see  that  each  assistant  enters  his  notes  in  the  proper  book  so 
neatly,  completely  and  correctly  that  at  the  end  of  any  day's 


CITY    SURVEYING.  187 

work  the  hook  ma}-  he  handed  to  any  other  assistant  who  would 
be  able  to  continue  the  work  without  the  least  possible  duplica- 
tion or  loss  of  time. 

ICach  assistant  should  be  required  to  carry  with  him  'the 
proper  note-book,  and  to  make  in  it  the  original  notes  of  the 
work.  If  this  is  done,  the  field-book  may  be  presented  as  evi- 
dence in  case  of  law  suits,  but  it  could  not  be  presented  as  evi- 
dence had  the  notes  been  copied  in  it  from  other  books  or  from 
scraps  of  paper. 

Note-books  should  not  be  permitted  to  litter  the  draughting 
tables  or  desks  of  the  office.  When  not  in  use  they  should  be  in 
their  proper  places  on  the  shelves,  or  in  cases. 

Kach  member  of  the  office  staff  should  be  impressed  with  the 
fact  that  surveys  are  expensive  and  that  the  data  contained  in 
these  note-hooks  arc  valuable.  Books  should  not  be  carelessly 
thrown  about,  but  on  the  contrary  should  be  carefully  pre- 
served and  everything  should  be  done  to  make  the  records 
readily  available  for  future  reference. 

185.  Detail  Maps. — Tn  addition  to  the  large  wall  map  of 
the  city  there  should  be  smaller  maps  to  a  larger  scale,  showing 
all  essential  details  of  lines,  angles,  monuments,  distances,  etc. 
The  wall  map  may  be  divided  into  sections  by  lines  at  right 
angles  to  each  other,'  or  by  streets  and  streams  into  sections 
corresponding  to  the  smaller  maps.  This  enables  the  detailed 
map  of  any  section  of  the  city  to  be  found  with  the  least  loss 
of  time  and  trouble.  On  these  detail  sheets  the  water,  gas, 
sewer,  and  steam  mains,  telephone  conduits,  etc.,"  should  be  rep- 
resented by  different  colored  inks  or  by  specially  dotted  lines.  If 
there  are  many  of  these  pipe  lines,  it  may  be  necessary  to  have 
several  copies  of  each  sheet,  one  devoted  exclusively  to  water 
service  (called  the  water-pipe  map),  one  to  sewerage,  etc. 

Such  maps  should  be  made  on  the  best  quality  of  mounted 
egg-shell  paper  and  should  be  service  maps  on  which  every 
change  in  pipe  lines  should  be  noted  immediately.  If  it  is  con- 
sidered necessary  to  have  records  of  conditions  at  different 
dates — i.  e.,  on  the  first  of  January  each  year — tracings  of  these 
service  sheets  may  be  made,  dated  and  filed. 


188  SURVEYOR'S  HAND   BOOK. 

An  excellent  plan  for  standard  sizes  for  drawings  is  to  acY-pt 
the  full  sheet,  half  sheet,  quarter  sheet,  and  eighth  sheet  plan, 
and  the  dimensions  of  these  can  be  for  full  sheets,  24x36  ins. ;  for 
half  sheets,  24x18  ins.;  for  quarter  sheets.  12x18  ins.;  and  for 
eighth  sheets,  12x9  ins.  Each  sheet  should  be  trimmed  %  to  1 
in.  outside  the  border  except  on  the  left,  where  a  double  margin 
should  be  left  for  binding  purposes. 

However,  it  is  useless  to  have  or  to  demand  accurate  city 
maps  and  drawings  and  not  at  the  same  time  provide  safe  and 
secure  repositories  for  such  records.  Substantial  cases  should  be 
constructed  with  a  set  of  drawers  (say  40x27  ins.  inside  dimen- 
sions) for  the  full  size  drawings.  For  the  half  size  drawings 
the  40  by  27  drawer  can  be  divided  by  a  thin  partition  across  the 
middle,  dividing  it  into  two  compartments  about  27x10%  ins. 
Another  set  can  be  provided  for  the  quarter  size  drawings 
where  the  40x27-in.  drawer  has  two  divisions  or  partitions  at 
right  angles  to  each  other ;  and  in  a  similar  way-  the  eighth  size 
drawings  can  be  provided  for.  The  drawers  should  be  numbered 
consecutively  and  if  divided  into  compartments  for  fractional 
sizes  each  compartment  should  be  given  a  letter  and  the  draw- 
ings in  it  numbered  in  a  special  place  on  the  drawing  in  addition 
to  the  general  number  that  it  must  bear.  Thus  the  drawing 
should  be  labeled  "Drawer  26  D,  Sheet  14,v  in  one  corner,  while 
the  general  number  76  will  indicate  that  it  is  the  76th  drawing 
made  by  the  city.  The  legend  "Drawer  26  D,  sheet  14,"  indi- 
cates that  it  is  to  be  replaced  in  drawer  26  in  compartment  D, 
between  sheets  13  and  15. 

In  addition,  a  systematic  record  should  be  kept  showing 
clearly  what  each  numbered  drawing  refers  to  in  the  general 
series.  An  alphabetical  list  should  be  made  of  these  drawings, 
where  the  leading  word  in  title  or  location  will  indicate  the 
character  of  the  drawing.  Better  than  this,  however,  is  a  card 
catalogue  where  every  map  is  cross-indexed  in  such  a  manner 
that  it  may  be  readily  found.  The  card  catalogue  has  many 
advantages  over  the  book  catalogue,  in  that  references  can  be 
made  with  greater  dispatch,  and  corrections  and  new  insertions 
can  be  made  without  disturbing  the  other  records. 


CITY    SURVEYING,  180 

186.  Orders,  Bids,  Etc. —  It  is  doubtful  whether  it  is 
necessary  to  mention  the  necessity  of  keeping  a  record  of  all 
correspondence,  orders,  etc.,  as  this  is  the  usual  practice  of  every 
good  business  man,  and  every  engineer  should  be  a  good  business 
man,  as  far  as  the  city  is  concerned  at  least. 

Contracts  and  specifications  are  important  documents  in  con- 
nection with  large  undertakings  or  important  works,  and  these 
should  be  kept  in  a  fire  proof  safe,  to  which  only  the  trusted 
members  of  the  staff  have  access.  Specifications,  results  of  tests, 
and  other  data  on  miscellaneous  matters  should  be  indexed  and 
may  be  filed  in  a  manner  similar  to  that  for  drawings. 

Bibliography. — "Theory  and  Practice  of  Surveying."  By 
the  late  J.  B.  Johnson.  This  book  has  long  been  a  standard 
work  for  the  surveyor  and  engineer.  Its  chapter  on  City  Sur- 
veying was  prepared  by  William  Bouton,  City  Engineer  of  St. 
Louis,  Mo.,  and  gives  the  conditions  necessary  for  high  grade, 
accurate  city  surveying. 

"Principles  and  Practice  of  Surveying."  By  Breed  and 
Hosmer.  An  excellent  book  for  the  city  engineer,  containing  full 
directions,  discussions,  and  illustrations  of  many  problems  that 
confront  the  city  engineer. 

"Engineering  Work  in  Towns  and  Cities."  By  Ernest  Mc- 
Cullough.  While  the  author  disclaims  any  intention  of  writing 
for  city  engineers  of  cities  over  10,000  population,  the  limit 
should  have  been  placed  at  50,000  instead  of  10,000.  The  book  is 
a  history  of  city  surveying.  With  gloves  off  it  deals  with  the 
qualifications  necessary  for  the  position  of  city  engineer,  the 
compensation  he  should  receive,  the  problems  he  has  to  solve, 
the  difficulties  he  has  to  meet,  how  to  keep  city  records,  the 
necessary  theory  and  principles  for  the  various  duties  of  the 
position,  including  the  location  of  monuments,  roads,  walks, 
pavements,  sanitation,  drainage,  sewerage,  water  supply,  con- 
crete, contracts  and  specifications,  office  system,  city  engineer's 
records  and  field  work.  It  ranks  as  possibly  the  best  book  be- 
fore the  public  for  the  use  of  the  city  engineer  and  especially  for 
that  city  engineer  who  wishes  to  learn  the  best  methods. 


CHAPTER  XI. 
PLOTTING  AND  LETTERING. 

187.  Plots. — After  a  farm  is   surveyed  a  line  map  of  the 
farm  or  land  should  be  made  to  some  convenient  scale,  for  the 
purpose  of  showing  the  shape  of  the  farm  or  body  of  land,  its 
connections  with  adjoining  properties,  and  its  location  with  re- 
spect to  natural  objects.     Such  a  plot  should  contain  the  follow- 
ing data : 

1.  Boundary  lines. 

2.  Bearing  and  distance  printed  on  each  line. 

3.  All  corners  described,  as   "a  hickory  1   ft.   diam.,  marked 
H,"  "a  stone." 

4.  Names  of  adjoining  property  owners. 
.">.     Meridian,  or  north  and  south  line. 

G.     Owner's  name  printed  inside  plot. 

7.  Number  of  acres  printed  under  owner's  name. 

8.  Complete  set  of  field  notes  printed  below  plot. 

Fig.  105  illustrates  the  plot,  description  and  style  of  letters. 

There  are  various  methods  used  in  making  a  plot  from  the 
field  notes.  These  are  generally  known  as  the  protractor,  the 
tangent,  the  sine,  or  the  co-ordinate  method. 

188.  Protractor    Method. — A    protractor,    Fig.    102,    is    a 
semicircle   of  horn,  celluloid,   German   silver,   etc.,   graduated   to 
half  degrees.     A  diameter  line  is  marked  at  one  end  0°  and  at 
the  other  end  180°.     A  bearing  is  laid  off  by  placing  the  center 
of   the   protractor    over   the   point    and    the    diameter   along   the 
meridian  and  the  protractor  to  the  right  or  left  of  the  meridian 
as  indicated  by  the  last  letter  of  the  bearing;   that  is,  east  for 
the  right  and  west  for  the  left.     A  point  is  made  on  the  circum- 
ference of  the  protractor  at  the  point  of  the  correct  bearing,  the 
protractor  is  moved   and  this  point  joined  by  a  line   to  the  be- 
ginning line  or  point.     The  length  of  the  course  is  then  laid  off 
on  this  line  to  the   scale  of  the  map.     Through   the  point  thus 
located  another  meridian  is  located  and  the  bearing  is  laid  off  as 
before. 

190 


PLOTTING    AND    LETTERIXG. 


191 


189.  Latitude  and  Departure  Method. — Begin  at  some 
point  A  as  in  Fig.  103,  and  lay  off  the  latitude  AB  due 
north  and  south  from  A,  and  through  the  point  thus  located 
draw  an  east  and  west  line  and 
lay  off  the  departure  on  this  \ 
line,  and  join  the  point  thus 
located  to  A.  Lay  off  the  lati- 
tude of  next  course  on  line 
through  C,  and  through  the 
point  thus  located  draw  an- 
other east  and  west  line  and 
lay  off  the  departure  on  this 
line,  thus  locating  the  point  D. 
Proceed  as  above  until  all  the 
points  are  located. 

190.  The  Tangent  Method. 
— To  lay  off  a  line  making  a 
given  angle  with  a  given  line 
at  a  given  point  A,  Fig.  103. 
by  the  tangent  method,  lay  off 
AB  equal  to  ten  parts  on  some 
scale,  and  at  B  erect  a  per- 
dicular  to  the  given  line,  and 
on  this  perpendicular  lay  off 
CB  equal  to  ten  times  the  nat- 
ural tangent  of  the  angle  de- 
sired; join  C  to  A.  Thus,  to 
lay  off  an  angle  of  29°  41',  we 
find  from  the  table  that  the 
natural  tangent  of  '29°  41'  — 
.5701).  Make  AB  equal  to  ten 
parts  and  lay  off  CB  equal  to 
5.7  parts,  thus  locating  C ;  then 
join  C  to  A  and  you  have  the 
angle  required. 

191.     The   Sine   Method. — To   lay    off  a    given   angle   at   a 
given  point  by  the  natural   sine  method,  take  a  radius  equal  to 


192 


SURVEYOR'S   HAND    BOOK. 


ten  parts  and  with  the  given  point  as  a  center  describe  a  circle. 
On  a  perpendicular  to  the  given  line  at  the  given  point  lay  off  A3, 
Fig.  104,  equal  to  ten  times  the  natural  sine  of  the  angle  required.. 
Through  3  draw  a  line  parallel  to  the  given  line  cutting  the  cir- 
cumference of  the  circle  at  B,  join  B  to  A  and  BAN  is  the  angle 
required.  Example :  To  construct  an  angle  of  33°  22'  we  find 
that  the  natural  sine  of  the  angle  33°  22'  is  .5500.  After  de- 
scribing the  circle  whose  radius  is  ten  parts,  lay  off  A3  equal  to 
5.5  parts,  and  draw  B3  parallel  to  the  line  AN  and  join  B, 
where  it  cuts  the  circumference  of  the  circle,  to  A,  and  BAN 
then  will  be  an  angle  of  33°  22'. 

192.     Co-ordinate  Method. — Plotting  can    be   done  by   the 
Co-ordinate  Method  :     Determine  the  co-ordinates  of  each  point 


Fig.    103. 


Fig.    104. 


with  respect  to  axes  (through  the  initial  point,  if  convenient) 
and  plot  from  .the  axes  each  time.  This  method  will  ?void 
carrying  forward  any  error,  as  each  corner  of  the  survey  is 
found  by  returning  to  the  original  axes.  The  Y  ordinate  of  any 
point  is  equal  to  the  sum  (algebraic)  of  the  latitude  of  the  pre- 
vious points  and  its  own  latitude.  The  X  ordinate  is  equal  to  the 
sum  of  the  previous  departures  plus  its  own.  Using  this 
table  of  corrected  latitudes  and  departures  insures  the  closing 
of  the  plot.  This  is  most  accurate  method  for  any  large  plot, 
as  previous  to  plotting  the  sheet  can  be  checked  off  in  squares 
accurately,  say  1,000  ft.  on  each  side,  and  table  of  ordinates 
computed,  etc. 


PLOTTING    AND    LETTERING. 


193 


Scale 


Beginning    at  a   stone  on   Bull  CreeK    a 
corner    to  R  A  Jones    and  John  Cusler 
Thence     with    Ouster's     line   S.4-I*E.  IOO  poles 
To  a   blacK  oaK    a  corner   To    John  CusTer  ond 
D.F?. Thomas  ,-  Thence    with  Thomas'    line 

SS9°W  f -41  poles   To    a    hicKovy    a  corner  To 
D.TR.ThornaS ;-    thence    with  ThomasMine 

N.G9*  W    99  poles    To  a   gum    on     Bull 
CreeK      o.    corner     To    JD.K.Thomas     onct 
TC.G  ore;- Th  ence     up  the    creeK    with  the 
meanders    of    the    Sanne     to   the    point   of 
beginning  ;-  containing     A-  S.9    acres. 


Fig.    105. 


194 


SURVEYOR'S  HAND   BOOK. 


193.  Correcting  the  Plot. — For  the  very  same  reasons 
that  the  latitudes  and  departures  very  rarely  balance,  the  plot 
when  completed  to  scale  will  very  rarely  close  by  an  amount 
equal  to  AA' ,  Fig.  106.  In  balancing  we  really  shift  each  corner 
in  the  direction  of  AA' ,  a  distance  proportional  to  its  length 
from  the  beginning  corner.  To  some  scale  lay  off  on  a  straight 
line  the  length  of  the  courses  ABCDA',  and  on  a  line  at  right 
angles  to  this  line  lay  off  AA'  and  through  the  points  B,  C,  and 
D  draw  parallels  to  AA'. 

Through  B,  C ,  and  D  on  the  plot  draw  lines  parallel  to  A  A' 
and  on  these  lines  lay  off  distances  equal  to  the  amount  of  cor- 
rection,      locating       the 

II  points  B' ,  C'  and  D'  in 

the  direction  that  A'  has 
to  be  moved  to  close. 
Then  connect  these 
points  and  close  the 
plot. 

194.  Lettering. — Ev- 
ery surveyor  or  engi- 
neer should,  learn  some 
one  system  of  free-hand 
letters,  similar  to  that  in 
Fig.  107,  or  some  other 
standard  system.  Many 
conclude  before  trial 
that  they  can  not  letter 

well,  or  even  make  a  decent  letter.  While  there  is  no 
royal  road  to  good  lettering,  it  is  possible  for  every 
surveyor  or  engineer,  not  afflicted  with  palsy  or  extreme  nervous- 
ness, to  learn  and  execute  a  good,  plain  system  of  letters.  But  it 
requires  care  and  implicit  obedience  to  rules.  Eternal  vigilance 
and  constant  practice  are  required  till  a  system  of  letters  is  once 
learned.  After  an  experience  of  over  twenty  years  in  teaching, 
it  can  be  asserted  that  the  special  books  on  lettering  are  far  supe- 
rior to  the  ordinary  alphabets  printed  as  an  appendix  to  works 
on  surveying.  If  the  young  engineer  will  get  "Lettering,"  by 


S 


Fig.    106. 
make 


PLOTTING    AND    LETTERING.  195 

C.  W.  Reinhardt,  published  by  D.  Van  Nostrand  Company,  New 
York,  and  will  follow  instructions  faithfully,  he  can,  without 
doubt,  become  a  good  letterer.  There  is  no  necessity  for  fancy 
letters  in  a  drawing,  as  neatness,  legibility,  and  clearness  are  the 
fundamental  requisites.  One  of  the  most  effective  systems  of 
lettering  is  shown  in  Fig.  107.  Guide  lines  should  always  be 
drawn  before  the  lettering  is  commenced  and  the  student  should 
adhere  strictly  to  rules. 

Bibliography. — "Lettering."     By  Chas.  W.  Reinhardt.  Pub- 
lished by  D.  Van  Nostrand  &  Co.    This  book  explains  in  a  clear 


abcdefqhijklmnopqrstuvwxyz. 
ABCDEFOHIJKLMNOPOR5TU 
V  WXYZ.  /  23456  78910.  CROSS  SECTION 
SECTION  Extended  Lettering  Ordinary 

Compressed  Type.  INTERSTATE  BRIDbE.  Spvrmult%°DiamJ'Rn 

a  bcdefghijklmnopqrstuvwxyz. 
ABCDEFGHIJ  KLMNOPQRSTUV 
WXYZ.  I  23456789 10.    Ordinary  Lettering 
Compressed,  NEW  YORK  CENTRAL 


Fig.    107. 

and  concise  manner  the  system  of  letters  devised  by  the  author 
and  shows  by  concrete  examples  how  each  letter  should  be 
formed  and  how  constructed.  In  addition  to  this  a  well  selected 
set  of  examples  of  title,  heading,  and  detail  lettering  is  given 

"Mechanical  Drawing."  By  F.  E.  Giesecke.  Part  I.  Pub- 
lished by  Eugene  Dietzgen  Company.  This  book  has  grown  out 
of  the  necessities  of  the  office  and  class  room  and  gives  an  ex- 
cellent system  of  free-hand  letters  for  detail  work  and  full  in- 
structions are  given  for  the  construction  of  each  letter.  This 
book  meets  all  the  demands  that  a  learner  of  lettering  can  make. 


CHAPTER  XII. 
GOVERNMENT  SURVEYING. 

195,  Radii  of  Parallels.  —  Government  lands  are  bounded 
by  meridians  and  parallels  of  latitude.  If  AB,  Fig.  108,  is  pare 
of  a  parallel  of  latitude,  its  latitude  is  the  arc  BO  or  the  angle 
BOQ,  which  we  will  call  L.  Let  the  radius  of  the  earth  he  R 
and  the  radius  of  the  parallel  be  r  or  BH.  Then  in  the  right 
triangle  OBH, 


That  is, 


Or       r  =  R  Cos  L  . 


196.  Angular  Convergence  of  Meridians. — The  two  merid- 
ians PA  and  PB,  Fig.  108,  at  the  points  A  and  B  have  the  direc- 
tion AK  and  BK,  respectively,  the  tangents  to  these  meridians. 
The  amount  of  convergence  is  the  angle  that  they  lack  of  being- 
parallel;  that  is,  the  angle  AKB  or  their  angle  of  intersection. 
Let  6  =  the  difference  of  longitude  of  A  and  B  =  angle  AHB  = 
EOQ.  In  the  sector  AHB  we  have: 

AB  =  -==-z  x  BH 

*)<  .o 

In  AKB  we  ha^e : 

AB  =  §£3  x  BK,  where  X  =  AKB. 
Consequently : 

X  6 


X  BK  =  -TO  X  BH 


57.3  f  «7.3 


BK 

X  =  e  »w.  L  ...... .,......,..,. (41) 

196 


GOVERNMENT  SURVEYING.  197 

197.     Linear    Convergence. — In    the    two    similar    sectors 
ABH  and  EOQ  we  have : 

AB  :  EQ  ::  BH   :  OQ 

BH 

AB  =  EQ   x  0g 


EQ  Cos  L 


Fig.    108, 


Fig.    109. 


If  DC  is  a  part  of  a  parallel  between  the  same  meridians  in 
latitude  L'  we  have : 

DC  =  EQCosL' 

Let  c  =  Convergence  ~  AB — DC 
—  EQCosL  —  EQCosE' 
=  EQ  (Cos  L  — Cos  L') 
DO       Cos  L' 
AB  ^UoTT: 
Therefore : 

AB(CosL—  CosL'.) 
c  =  AB-DC  =  z^j- ,.-'..;.  (42) 

Generally  we  do  not  know  the  difference  of  longitude  of  A 
and  B,  but  know  the  length  of  AB  in  miles,  and  it  is  nec- 
essary to  find  9  from  the  data  given.  The  length  of  one  degree 
on  the  equator  is  69,16  miles. 


198  SURVEYOR'S  HAND   BOOK, 

If  D  =  length  of  AB  in  miles,  then 

D 

AB  in  degrees  =  69.16  Cos  L 

But  X  =  9    Sin  L 

D  Sin  L 


Therefore  A  =  H 


69.16  Cos  L 


X"=  gg-jg  D  X  tan  L=52.05  D  tan  L 

198.  Off-Sets.  —  If  we  set  the  transit  at  B,  Fig.  109,  and 
set  the  zero  on  the  meridian  and  turn  off  a  right  angle  from 
this  meridian,  this  last  line  will  cut  to  the  left  of  A.  Draw  the 
sector  AKB  as  in  the  figure  and  make  the  angle  KBR  equal  to 
90°.  The  amount  the  line  BR  misses  A  is  called  the  off-set 

The  angle  ABR:==  one-half  X. 


IfAB=D,  we  have: 

AR^Off-srt-^-fir^  XD 

But  X  =  B  Sin  L  =  Q(j  16  tan  L 

W  D 

Therefore,          AR  =  tan  L  x 


This  is  the  off-set  in  miles.      If  D  is  in  miles  and  we  wish  the 
off-set  it    feet,  we  have  : 

D2 
Off-set=  69.16X57.3X2  ian  L  x  fc80 

Therefore  off-set  =  59  16x57  3x2  tan  LxD* 
=  .66618  tan  L  D* 
=  f  D2  /aw  L,  w^arZ^  .............  (43) 

199.     Running  Parallels.—  It   is  impossible  to  run   out  the 
parallel  of  latitude  with  the  transit  directly.     We  can  locate  the 


GOVERNMENT  SURVEYING.  199 

secant  BA  or  the  tangent  BR,  and  then  take  off-sets  to  the  curve 
of  latitudes  at  different  points,  which  are  generally  one-half 
mile  apart.  There  are  two  methods  of  locating  points  on  the 
parallel  of  latitude,  the  secant  method  and  the  tangent  method. 

200.  Tangent  Method.— Set  up  the  transit  at  B  and  sight 
along  the  meridian  BK.     Then  turn  off  an  angle  K'BR  equal  to 
90°.    The  line  of  sight  will  now  locate  the  line  BR,  which  is  tan- 
gent to  the  latitude  curve.     To  obtain  the  off-sets  from  this  tan- 
gent line  to  the  curve  at  any  point  on  BR,  let  d  —  distance  from 
the  point  to  B.    Then  we  have  from  formula  (43)  : 

Off-SCt  =%<f   tan     L 

After  the  full  distance  has  been  measured,  the  point  R  is 
located.  To  locate  the  point  A,  set  up  the  transit  at  R,  and 
sight  along  the  line  BR,  and  then  turn  off  an  angle  of  90° — X°. 
Tfre  line  of  sight  will  now  locate  the  meridian  RAK,  and  if  we 
measure  the  distance  RA  along  this  line 
an  amount  equal  to  the  off-set,  it  will  lo- 
cate the  point  A  on  the  parallel  of  latitude 
passing  through  B. 

201.  Secant     Method. — Set     up     the 
transit  at  B,  as  before,  and  sight  along  the 
meridian   BK;   then   turn   off  an    angle   of 
00° — VzX.     The  line  of  sight  will  now  lo- 
cate the  secant   line  BA,  which   can  be  run  out  to  the  distance" 
BA.      To    locate    points    on    the   parallel    of    latitude    for    either 
method,  off-sets  must  be  taken  from  the  tangent  or  secant. 

202.  Intermediate    Off-Sets. — To    find    the  off-sets  at  any 
intermediate  point  between  B  and  A,  let  d  —  distance  BT  or  BS, 
Fig.  110.     The  point  C  on  the  curve  can  be  located  by  the  off-set 
TC  from  the   tangent  or  the   off-set  SC  from  the   secant.      The 
angle  SBT=%  X°. 

.;  .  Secant-tangent  off-set  ST  =  %  X°  BT  •*-  57.3, 

But  X  =  Q$-fttan.L. 

D  Dd 

ST  ^  tanLd  +  57-3  r  2x69.16x57.3^  L' 


200  SURVEYOR'S  HAND    BOOK. 

If  ST  is  in  feet  and  D  and  d  are  in  miles,  then 
Secant-tangent  off-set,  ST=%Dd  tan.  L. 

To  find  the  off-set  from  the  tangent  BR  to  the  curve,  we  have, 
BT-—TC  (2CK  +  TC)  =  2CK  X  TC  +  TC: 

The  last  term  is  so  small  in  comparison  with  the  others  that  it 

can  be- omitted. 

Br*  =  2  TC  X  CK  =  2TC  X  BK  /.  TC  =  ~^ 

But  BK  =  R  Cot.  L  and  BT  =  d,  then, 

d2  d* 

TC  =  o  r>  r  f — r—^T&tunL.      If  the  offset  is  in  feet 
2  K  Cot.  L,      2K 

and  d  in  miles  we  have, 

Offset   TC=%d2  tan.  L. 

The  secant-curve  off-set  can  be  found  by  subtracting  the  tangent- 
curve  off-set  from  the  secant-tangent  off-set. 

•  '.SC=%Dd  tan.  L.  —  %  d'2  tan.L  =  2/3  d  (D  —  d)  tan.L. 
The  secant-curve  off-set  is  equal  to  two-thirds  of  the  tangent  of 
the  latitude  multiplied  by  the  segments  into  which  S  divides  AB. 

203.  Example. — If  a  line  AB  is  six  miles  in  length  and  is 
a  parallel  of   latitude   where   L  =  45°    the   different   off-sets    for 
each  mile  can  be  found  as  follows : 

A.    Tangent-curve   off -set  =  %  d~  tan.  L.  =  %  d2  tun  45  =  %  d~. 

E.    Secant-tangent    off-set  =  %  Dd  tan.  L  =  %Dd  tan  45  =  % 
Dd. 

C.  Secant-curve  off -set  =  %  d  (D  —  d}  tan.  45  —  %  d  (D  —  d). 

—Off-sets.— 
Distance  d.         Secant-tangent.       Tangent-curve.         Secant-curve. 

1  4  .667  3.333 

2  8  2.667  5.333 

3  12  6.000  6.000 

4  16  10.667  5.333 

5  20  16.667  3.333 

6  24  24.000  0.000 
PROBLEM  75. — Fill  out  a  similar  table  when  latitude  =  36°. 

204.  Reference    Meridians    and    Standard    Parallels. — In 
those  states   where  public  lands   were  surveyed  by  government 
surveyors,  meridians  were  located  very  accurately  at  certain  in- 


G  O  VBRNMEN  T  S  UR  VE  YIXG. 


201 


tervals  and  parallels  of  latitude  were  also  accurately  located  at 
certain  distances  apart.  As  an  example  the  two  meridians  BC 
and  AD,  Fig.  Ill,  called  "reference  meridians,"  were  located  24 
miles  apart,  and  the  "Standard  Parallels,"  AB  and  CD,  were 
also  located  24  miles  apart.  This  makes  a  spherical  trapezoid 
whose  sides  are  nearly  24  miles  each.  The  six-mile  points  on 
these  sides  are  marked  and  joined  by  meridians  and  parallels, 
thus  dividing  the  area  into  smaller  trapezoids,  with  sides  6  miles 
each  way  approximately.  These  trapezoids  are  called  "Town- 
ships." 


x 

w 

z 

Y 

X 

Fig.    111.  Fig.    112. 

The  south  base  of  a  trapezoid  is  24  miles  on  a  standard 
parallel  and  the  next  standard  parallel  is  24  miles  to  the  north. 
If  the  latitude  of  the  south  base  is  40°,  find  the  amount  of  con- 
vergence. 

To  find  L'  the  latitude  of  the  north  base  we  have : 

One  degree  —  69. 16  miles. 

24 
24  miles  =      ~     degrees 


69.16 


=40°20'  49" 


24  [Cos  40°— Cos  (40°  20'  49")] 
c=  convergence  -  Cos  40° 

24(. 76604 -.76549) 

.76604  --12219  miles 

PROBLEM  76. — A  trapezoid  is  24  miles  each  way,  and  the  lati- 
tude of  the  mid-parallel  is  46°  30'.  Find  the  amount  of  converg- 
ence, 


202 


SURVEYOR'S  HAND   BOOK. 


PROBLEM  77. — Find  the  convergence  of  a  trapezoid  with  6-mile 
base  and  24  miles  north  and  south,  if  the  latitude  of  the  south 
base  is  36°. 

205.  Ranges.— In  each  State  or  Territory  a  principal 
meridian  was  located  as  BC,  Fig.  112.  It  received  a  name  clue 
to  some  locality,  as  the  Fayetteville  or  Butte  meridian.  Also  a 
principal  parallel  is  located  as  AB.  The  country  is  then  divided 
into  townships  on  either  side  of  these  axes  and  they  serve  as  co- 
ordinates in  locating  the  townships.  Thus  in  the  figure  all 
ranges  are  west  and  north.  Any  row  of  townships  running 


6 

5 

4 

5 

2 

1 

7 

a 

9 

JO 

n 

12 

J& 

17 

It. 

75 

74 

73 

73 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

37 

32 

33 

34 

35 

36 

Fig.    113. 

north  and  south  is  called  a  Range,  while  that  running  east  and 
west  is  called  a  Tier.  Each  township  is  defined  as  in 
Range  1,  2,  3,  or  4,  Tier  1,  2,  3,  or  4,  as  the  case  may  be,  number- 
ing from  the  Principal  Meridian  and  Principal  Parallel.  Thus 
the  township  crossed  will  be  Range  3  west,  Tier  4  north. 

206.  Townships. — The  trapezoid  in  Fig.  112,  24  miles  each 
way,  was  surveyed  between  reference  meridians  and  standard 
parallels.  If  the  six-mile  points  on  the  north  and  south  lines 
are  marked  the  spherical  trapezoid  would  be  divided  into  ap- 
proximate squares  six  miles  each  way,  called  townships. 


GOVERNMENT  SURVEYING.  -jo:i 

Each  township  is  divided  into  30  approximate  squares,  about 
one  mile  on  each  side,  called  sections.  The  sections  in  each 
township  are  numbered  as  shown  in  Fig.  113.  Section  number  1 
is  in  the  northeast  corner  of  the  township,  while  number  30  is 
in  the  southeast  corner. 

207.  Dividing  Up  a  Township. — All  township  lines  on  the 
south  base,  on  the  standard  parallels,  are  full  G  miles,  as  are  all 
township  lines  on  the  meridians.  In  the  range  of  townships 
EBCF,  Fig.  112,  there  would  really  be  only  one  east  and  west 
line  that  was  fully  0  miles  long,  as  all  the  others  are  reduced  by 
the  convergence.  In  dividing  the  first  township  X  into  sections 
we  mark  off  full  miles  on  the  south  base  EB,  80  chains  each,  and 
also  full  miles  on  the  north  and  south  lines  BC  and  EF.  If  we 
made  the  north  and  south  division  lines  true  meridians  the  sec- 
tions would  decrease  materially  in  size  as  we  proceeded  north. 
To  counteract  this  and  keep  them  approximately  1  mile  each  way 
we  make  the  south  base  of  each  section  bordering  on  the  town- 
ship lines  80  chains  as  far  as  possible.  On  each  east  and  west 
township  line  we  commence  at  the  meridian  on  the  east 
side  of  the  township  and  measure  off  5  full  miles,  marking  the 
corners ;  thus  out  of  144  sections  in  a  Range  we  would  have  21 
sections  with  a  full  mile  for  the  south  base  instead  of  0  sections 
if  it  were  divided  by  true  meridians  on  the  mile  points  of  the 
standard  parallel  AB.  The  amount  of  convergence  of  the  town- 
ships X,  Y ,  Z}  and  W  will  be  practically  the  same  if  they  have 
equal  south  bases.  On  the  outlines  of  the  townships  the  corners 
are  marked  with  stones  or  posts  as  indicated  by  the  small  circles 
in  Fig.  113.  On  the  township  lines  the  full  mile  points  are  all 
established  and  marked  by  corners.  In  making  the  survey  of 
sub-division,  we  begin  on  the  south  base  of  the  township  at  cor- 
ner to  sections  35  and  30,  and  then  run  the  line  between  sections 
35  and  36  so  that  it  will  be  parallel  to  the  east  line  of  30.  In  the 
same  way  all  the  north-south  lines  are  run  parallel  to  the  east 
line  of  the  township  except  for  sections  from  1  to  0  inclusive. 
From  the  corner  of  1,  2,  11,  and  12  the  line  between  1  and  2  is 
run  directly  to  the  established  corner  on  the  north  base  of  the 
township.  The  lines  between  sections  2  and  3,  3  and  4,  4  and  5, 


204 


SURVEYOR'S  HAND   BOOK. 


and  5  and  6  are  run  in  a  similar  way.  On  all  north-south  lines 
five  full  miles  are  measured  from  the  south  base  of  township,  set- 
ting a  post  or  stone  at  the  end  of  each  full  mile  for  a  section 
corner.  The  east-west  lines  join  the  corners  on  the  north-south 
lines.  A  random  line  is  first  run  from  the  section  corners  to 
the  eastward  and  if  it  does  not  hit  the  corner,  the  correction  is 


Fig.    114. 

made  and  the  true  line  run.  The  east-west  lines  of  sections  31, 
30,  19,  18,  7,  and  6  receive  practically  all  the  effect  of  conver- 
gence of  the  township ;  and,  if  these  sections  are  divided  into 
quarter  sections,  the  shortage  in  length  is  thrown  into  the  west 
halves. 


GOVERNMENT  SURVEYING. 


205 


The  township  is  subdivided,  Fig.   114,  as  follows: 

Beginning  at  corner  1-2-35-36  on  the  south  base, 
thence  Nl'W  between  sections  35  and  36. 

Wire  fence,  bears  E.  and  W. 

Scattering  cottonwood  bears  east  and  west.  F.  G. 
Alexander's  house  bears  N28°W. 

Leave  cottonwood  timber  bears  east  and  west.  Enter 
road  bears  north. 

Southeast  corner  Alexander's  field.  Thence  along 
west  side  of  road. 

Cross  roads.     Bears  east  to  Mound  City. 
Bears  north  to  Link  City. 

Quarter  section  corner  point  falls  in  the  road. 

Enter   dense  cottonwood  timber;  bears  N54°E. 

Set  locust  post  4"x4" —  2'  in  the  ground  for  corner- 
sections  25,  26,  35,  and  36. 

Thence  S89°  57'E  on  a  random  line  between  sections: 
25  and  36. 

Set  temporary  quarter  section  corner  post. 

Intersect  east  line  of  township  3  links  north  of  corner 
of  sections  25,  30,  31,  and  36,  which  is  a  sandstone 
5"x8"  set  5"  above  the  ground,  marked  and  wit- 
nessed. 

Thence  N89°56'W  on  a  true  litre  between  sections  25 
and  36 'Over  level  bottom  land. 

Cherry  Creek,  12  links  wide,  clear  water,  1  ft.  deep, 
gentle  current,  sandy  bottom,  course  northwest. 

Heavy  timber,  bears  north  and  south. 

Leave  heavy  timber  bearing  north  and  south. 

Deposit  a  quart  of  charcoal  12  ins.  in  the  ground  as  a 
quarter  section  corner.  Dig  pits  18x18x12  ins.  east 
and  west  4  ft.  and  raised  a  mound  of  earth  3V&  ft 
base  by  Wz  ft.  high  over  the  deposit. 

Enter  heavy  timber  bears  north  and  south. 

Leave  heavy  timber,  enter  scattering  timber  bears 
N25°E. 

Comer  sections  25,  26,  35,  and  36. 


206  SURVEYOR'S  HAND   BOOK. 

Thence  Nl'W  between  sections  25  and  26. 

25.36  Right    bank    of   Yellowstone    river.      Set    locust  post 

4"x4"  —  24"  in  the  ground  for  meander  corner  for 
sections    25    and   26,    marked    MC    on    north    side. 
Entered  shallow  water  1  to  2  ft.  deep. 
Across  shallow  channel  64  links  wide  to  sand  bar. 
32.12  To  right  bank  of  main  channel,  course  east. 

40  Quarter  section  corner  falls  in  the  river. 

49.46  Left  bank  of  Yellowstone  river,  12  ft.  high,  deposited 

a  marked  stone  12  ins.  in  the  ground. 
55.70  Wire  fence  bears  east  and  west. 

62.80  Telegraph  line  bears  east  and  west. 

Set  cedar  post  for  corner  sections  23,  24,  25,  and  26. 


40 
79.98 


39.99 


58.00 
79.98 


Thence  S89°56'E  on  a  random  line. 

Set  temporary  quarter  section  corner. 

Intersect  east  line  of  township  3  links  north  of  section 
corners  25,  24,  30,  and  19;  which  is  a  sandstone 
5x9  ins. — 4  ins.  above  ground  marked  and  witnessed. 

Thence  back  N89°55'W  on  a  true  line  between  sec- 
tions 25  and  24. 

Set  a  cedar  post  3  ft.  by  3  ins.  square  with  a  marked 
stone  24  ins.  in  the  ground  for  a  quarter  section 
corner. 

Short  creek,  3  links  wide. 

Cor.  of  sees.  23,  24,  25,  and  26. 


The  survey  progresses  in  this  way  till  we  reach  the  corner  of 
sections  1,  2,  11,  and  12,  when  we  continue  as  follows: 

Beginning  at  corner  1,  2,  11,  12. 

Thence   Nl'W  on   a  random  line  between  sections  1 

and  2. 

40  Set  temporary  quarter  section  corner. 

79.77  Intersect  north  line  of  township  at  corner  of  sections 

1,   2,  35,   and  36,   which   is  a  limestone   6"x6" —  5' 
above  ground,  marked  and  witnessed. 
Thence  Sl'E  on  a  true  line  between  sections  1  and  2 
39.77  Set  marked  stone  for  quarter  section  corner. 


In  the  next  Range  of  sections  we  begin  at  corner  on  south  base 
2,  3,  34,  and  35,  and  proceed  as  before.  In  this  case,  after, the 
surveyor -has  located  the  corner  2,  3,  10,  11  he  runs  a  random 
line  N.  2'  W.  between  sections  2  and  3  and  misses  the  corner  ofi 


GOVERNMENT  SURVEYING.  207 

sections  2,  3,  34,  and  35,  five  links  to  the  west,  and  thence  runs  due 
south  on  a  true  line  between  sections  2  and  3. 

Bibliography. — "A  Manual  of  Land  Surveying."  By  F. 
Hodgman.  374  pages.  A  very  valuable  book  for  the  surveyor 
or  field  engineer  in  surveying  the  public  lands.  A  unique  and 
very  important  feature  is  a  digest  of  the  legal  decisions  ,by  the 
different  State  and  Federal  courts  in  regard  to  U.  S.  Lands,  sur- 
veys, conflicts,  etc. 

"A  Manual  of  Surveying  Instructions."  Prepared  under  di- 
rection of  the  Commissioner  of  the  General  Land  Office  of  the 
United  States,  Washington,  D.  C.  It  contains  full  and  minute 
directions  for  the  execution  of  surveys  in  the  field  in  conformity 
to  the  laws  of  the  United  States. 


CHAPTER  XIII. 

TRIGONOMETRIC   FORMULAS. 

208.  Formulas  for  Right  Triangle. — In  the  right  triangle 
ABC,  Fig.  115,  where  C  is  the  right  angle,  and  a,  b,  and  c  are 
the  sides,  we  have  the  following  expressions  for  the  different 
trigonometric  functions : 


Fig.    115. 


Fig.    116. 


.       w 

sin  A=—; 


cos  A=—, 


tan  A=-j-; 


esc  A=-, 


sec  A=r-j-; 


cot  A=~; 


Also, 


sin  A— 


cos  A 


tan  A-- 


__ 

cot  A 


esc  A'  sec 

The  following  relations  are  sometimes  useful : 
sin*  A  +  cos*A=l; 

1  +  tanzA  =  seczA; 
1  +  cot*  A  =  csc*A; 

209.  Solutions  for  Right  Triangle. — There  are  four  gen- 
eral cases  that  can  occur,  according  to  the  data  given,  which 
may  be — 

I.     The  hypotenuse  and  one  leg ; 
II.     The  two  legs; 

III.  The  hypotenuse  and  one  of  the  acute  angles ; 

IV.  A  leg  and  an  acute  angle. 

The  data  given,  the  data  required,  and  the  solutions  are  given 
in  the  following  tabular  statement.  It  is  assumed  that  if  angle 
B  is  known,  A  is  also  known. 

208 


TRIGONOMETRIC    FORMULAS.  209 

Given          Required  Solutions 

a,  c b,A,B  sinA=—>    b  =  c  cos  A  ;  B  =  90  —  A. 

<*,b A,c,B  tanA=y;    c=  --^p     B  =  90  —  A. 

c,  A a,  b,  B  a  =  c  sin  A;   b=c  cos  A  ;  B  =  90  —  A . 

f.£->  ^-^T;    b=~    B  =  W-A. 


210.     Oblique    Triangle. — In    the    general    triangle    ABC, 
Fig.    110,   threq  parts,  one  of   which  must  be  a   side,  have  to  be 
given    to   find    the    other    three.      There    are    four   general   cases 
according  to  the  data  given.     Thus  we  may  have : 
I.     Two  angles  and  the  included  side ; 
II.     Two  sides  and  the  included  angle; 

III.  Three  sides  ; 

IV.  Two  sides  and  an  angle  opposite  one  of  them. 

The   given    parts,    the    required    parts,   and   the    formulas   for 
solution  are  given  in  the  following  table : 

Given      Required.  Formulas  for  Solutions 

A,C,b     B,c,a      ^180 

b,  c,  A      B,  C,  a      B  +  C=180—A ;  Ian  %(B-C)  =    ~ 
b  sin  A 


=    \  fiz^itf) 

\ 


5  (5-a) 
C/t«?cfe  :  ,4  " 

6  s/;/  .  1 


A     B,C,c         sin 


210  SURVEYOR'S  HAND   BOOK. 

a  sin  C 
c=s~wTA" 

Case  IV  is  sometimes  ambiguous.  We  may  have  the  follow- 
ing conditions  and  results  : 

If  A  is  obtuse,  and  a>b  there-is  one  solution; 

If  A  is  acute  and  a  =  or  >  b,  there  is  one  solution; 

If  A  is  acute  and  a<b  and  a>b  sin  A,  there  are  two  solutions; 

If  A  is  acute  and  a  <  b  and  a=b  sin  A  ,  there  is  one  solution  ; 

If  A  is  acute  and  a<b  and  a<b  sin  A,  there  is  no  solution. 

211.  Right  Spherical  Triangle.  —  If  ABC  is  a  right  spheri- 
cal triangle  where  (7  —  90°,  and  the  hypotenuse  (c),  and  the  two 
acute  angles   (A  and  B)  are  treated  as  co-parts,  the  five  parts  of 
the  triangle  in  order  are  a,  b,  90  —  A,  90  —  c,  and  90  —  5.     To 
these  five  parts  the  following  laws  (discovered  by  Napier)  apply: 

Tangent  Law  :  The  sine  of  any  part  is  equal  to  the  product 
of  the  Tangents  of  the  Adjacent  parts. 

Cosine  Law  :  The  sine  of  any  part  is  equal  to  the  product 
of  the  Cosines  of  the  Opposite  parts. 

The  right  angle  C  is  not  counted  or  regarded  as  a  part  and  a 
and  b  are  regarded  as  adjacent  parts  as  no  significant  part  comes 
between  them.  For  any  one  part  the  two  adjacent  parts  are 
those  next  to  it,  while  the  opposite  parts  are  the  other  two,  or 
parts  once  removed  from  the  special  part  under  consideration. 
Thus  for  90  —  A,  the  adjacent  parts  are  b,  and  90  —  c,  while 
the  opposite  parts  are  a  and  90  —  B. 

By  the  application  of  Napier's  laws  we  can  solve  any  spherical 
triangle  where  the  three  given  parts  are  two  sides  and  an  angle 
or  two  angles  and  a  side. 

212.  Oblique    Spherical    Triangle.  —  If    three    sides    of    a 
spherical  triangle  ABC,  are  given,  let 

2s  =  a  +  b  +  c,  and  we  have, 

sin  \A  =      \sin(s~b}sin(s-c} 
\  sin  b  sin  c 


tan 


s^n  s  s^n  (s—a) 

If  the  three  angles  are  given,  pass  to  the  polar  triangle  and 
solve,  and  then  pass  back. 


CHAPTER  XIV. 

TABLES  OF 

LOGARITHMS  OF  NUMBERS. 
LOGARITHMIC     SINES,     COSINES,     TANGENTS,     AN 

COTANGENTS. 

NATURAL  SINES  AND  COSINES. 

NATURAL  TANGENTS   AND   COTANGENTS. 

CUBIC  YARDS  PER  100  FT.  FOR  VARIOUS  SLOPES. 


211 


212 


TABLE    I.      LOGARITHMS    OF    NUMBERS. 


No. 
100 

V 

000000 

A 

000434 

000868 

001301  1001734 

002166 

002598 

003029 

003461 

9 

003891 

Dlff 
132 

1 

4321 

4751 

5181 

5609:  6038 

6466 

6894 

7321 

7748 

8174 

428 

a 

8600 

jk)26 

9451 

9876010300 

010724 

011147 

011570 

011993 

012416 

424 

3 

012837 

01325'J 

0136SO 

014100J  4521 

4940 

5360 

5779  !  6197 

6616 

420 

4 

7033 

7451 

7868 

82841  8700 

9116 

9532 

994  7  i  020361 

020775 

416 

6 

021189 

021603 

022016 

022428022841 

023252 

023664 

024075  4486 

4896 

412 

6 

5306 

5715   6125 

6533  6942 

7350 

7757 

8164!  8571 

8978 

40S 

7 

9384 

9789 

030195,030600031004 

031408 

031812 

032216:032619 

033021 

404 

8 

033424 

033826 

4227  4628 

5029 

5430 

5830 

6230   6629 

7028 

400 

9 

7426 

7825 

8223  8620 

9017 

9414 

9811 

040207 

040602 

040998 

397 

110 

041393 

041787 

042  1  82!  042576 

042969 

043362 

043755 

044148 

044540 

044932 

393 

1 

6323 

5714 

61051  6495 

6885 

7275 

7664 

8053 

8442 

8830 

390 

2 

9218 

9606 

9993  050380 

050766 

051153 

051538 

051924 

052309 

052694 

386 

3  053078 

053463 

053846  4230 

4613 

4996 

5378 

5760 

6142 

6524 

383 

41  6905 

7286 

7666  8046 

8426 

8805 

9185 

9563 

9942 

06032<l 

379 

6 

060698 

061075 

061452061829 

062206 

062582 

062958 

063333 

063709 

4083 

376 

6 

4458 

4832 

5206i  5580 

5953 

6326 

6699 

707l|  7443 

7815 

373 

7 

8 

8186 
071882 

8557 
072250 

8928  9298 
072617072985 

'  9668 
073352 

070038 
3718 

070407 
4085 

070776071145 
4451   4816 

071514 
5182 

370 
366 

9 

6547 

6912 

6276  6640 

7004 

7368 

7731 

8094 

8457 

8819 

363 

120 

079181  079543 
082785083144 

0799041080266 
083503  3861 

080626 
4219 

080987 
4576 

081347 
4934 

081707 
5291 

082067 
5647 

082426 

6004 

360 
357 

2 

63601  6716 

7071   7426 

7781 

8136 

8490 

8845 

9198 

9552 

356 

3 

9905 

090258 

090611  090963 

091315 

091667 

092018 

092370 

092721 

093071 

352 

4 

093422 

3772 

4122  4471 

4820 

5169 

5518 

5866 

6215 

6562 

349 

6 

6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

100026 

346 

6 

100371 

100715 

101059 

101403 

101747 

102091 

102434 

102777 

103119 

3462 

343 

T 

3804 

4146 

4487 

4828 

6169 

6510 

6851 

6191 

6631 

6871 

341 

t 

7210 

7649 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

110253 

338 

9 

110690 

110926 

111263 

111599 

111934 

112270 

112605 

112940 

113275 

3609 

335 

130 

113943 

114277 

114811 

114944 

115278 

115611 

115943 

116276 

116608 

116940 

333 

1 

7271 

7603 

7934 

8265 

8595 

8926 

9256 

9586 

9915 

120245 

330 

2 

120574 

120903 

121231 

121560 

121888 

122216 

122544 

122871 

123198 

3525 

328 

3 

3852 

4178 

4504 

4830 

6156 

5481 

5806 

6131 

6456 

6781 

325 

4 

7106 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

969(1 

130012 

323 

6 

130334 

130655 

130977 

131293 

131619 

131939 

132260 

132580 

132900 

3219 

321 

6 

3539 

3858 

4177 

4496 

4814 

5133 

6451 

5769 

6086 

6403 

318 

7 

6721 

7037 

7354 

7671 

7987 

8303 

8618 

8934 

9249 

9564 

316 

8 

9879 

140194 

140503 

140822 

141136 

141450 

141763 

142076 

142389 

142702 

314 

9 

143015 

3327 

3639 

3951 

4263 

4574 

4885 

5196 

5507 

6818 

311 

140 

146128 

146438 

146748 

147058 

147367 

147676 

147985 

148294 

148603 

148911 

309 

1 

9219 

9527 

9835 

150142 

150449 

150766 

151063 

151370 

151676 

151982 

307 

2 

152288 

152594 

152900 

3205 

3510 

3815 

4120 

4424 

4728 

5032 

305 

3 

5336 

5640 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8061 

303 

4 

8362 

8664 

8965 

9266 

9567 

9868 

160168 

160469 

160769 

161068 

301 

6 

161368 

161667 

1619671162266 

162564 

162863 

3161 

3460 

3758 

4055 

299 

6 

4353 

4650 

4947J  5244 

5541 

5838 

6134 

6430 

6726 

7022 

297 

7 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9674 

9968 

295 

8 

170262 

170555 

170848 

171141 

171434 

171726 

1721119 

172311 

172603 

172895 

293 

9 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

6222 

6512 

5802 

291 

150 

176091 

176381 

176670 

176959 

177248 

177536 

177825 

178113 

178401 

178689 

289 

1 

8977 

9264 

9552;  9839 

180126 

180413 

180699 

180986 

181272 

181558 

287 

2 

181844 

182129 

182415  182700 

2985 

3270 

3555 

3R39 

4123 

4407 

285 

3 

4C91 

4975 

5259!  5542 

5825 

6108 

6391 

6674 

6956 

7239 

283 

4 

7521 

7803 

8084   8366 

8647 

8928 

9209 

9490 

9771 

190051 

281 

6 

190332 

190612 

190892191171 

191451 

191730 

192010 

192289 

192567 

2846 

279 

6 

3125 

3403 

3681   3959 

4237 

4514 

4792 

5069 

5346 

6623 

278 

7 

5900 

6176 

6453  6729 

7005 

7281 

7556 

7832 

8107 

8382 

276 

8 

8657  8932 

9206   9481 

9755 

200029 

200303 

200577 

200850 

201124 

274 

9 

201397 

201670 

201943 

202216 

202488 

2761 

3033 

3305 

3577 

3848 

272 

I* 

0 

1 

a 

3 

4 

5 

6 

7 

8 

9 

Dlff. 

TABLE    I.      LOGARITHMS    OF    NUMBERS. 


213 


HI 

0  I  1 

934 

5 

6 

7 

8 

9  iDltf. 

IftO! 

2041^!  204391 

204663204934205204 

205475 

205746  206016  206286 

206566 

271 

2 

6826:  7096 
9515   9783 

7365   7634   7904 
210051  210319:210586 

3173 
210353 

8441!  8710i  8979 
2111211211388  211654 

9247 
211921 

269 
267 

3J212188|212454 
4   48-14   5109 

2720   2986   3252 
5373   5633   590-2 

3513 
6166 

3783  4049   4314 
6430  6694  6957 

4579 
7221 

266 

264 

5i 

7484!  7747 

8010   8273   8536 

3798 

9060  9323  9585 

9846 

262 

0 

220103220370 

220631  22089-2  221153 

221414 

221675 

22  1936!  '222  196 

222456 

261 

7 

271  6!  2976 

3236   3496]  3755 

4015 

4274 

4533 

4792 

5051 

259 

8 

5309 

5568 

5826  6084 

6342 

6600 

6858 

7115 

7372 

7630 

258 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

230193 

256 

1 
170  230449 

230704 

230960231215 

231470 

231724 

231979 

232234 

232488 

232742 

255 

1 

2996   32.50 

3504  i  3757 

4011 

4264 

4517 

4770 

5023 

5276 

253 

2 

5528   5731 

60331  6285 

6537 

6789 

7041 

7292 

7544 

7795 

252 

3 

8046   8297 

8548  8799 

9049 

9299 

9550 

9800 

240050 

240300 

250 

4 

240549]  240799 

241048!241297 

241546 

241795 

242044 

242293 

2541 

2790 

249 

5 

3038)  3286 

3534   3782 

4030 

4277 

4525 

4772 

5019 

5266 

218 

6 

5513  5759 

6006  6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

7 

79731  3219 

8464 

8709 

8954 

9193 

9443 

9687 

9932 

250176 

245 

8250420|250664 

250903j25115l 

251395 

251633 

251881 

252125 

252368 

2610 

243 

9 

2353  3096 

3388  3580 

3822 

4064 

4306 

4548 

4790 

5031 

242 

180 

255273  255514 

255755I255996 

256237 

256477 

256718 

256958 

257198 

257439 

241 

1 

7679   7913 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

2  '26007  1 

260310 

260548 

260787 

261025 

261263 

261501 

261739 

261976 

262214 

238 

3 

2451 

2638 

2925|  3162 

3399 

3636 

3873 

4109 

4346 

4582 

237 

4 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

6 

7172   7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

6 

9513!  9746 

9980 

270213 

270446 

270679 

270912 

271144 

271377 

271609 

233 

7i271842  272074 

272306 

2538 

2770 

3001 

3233 

3464 

3696 

3927 

232 

8 

4158 

4339 

4620 

4850 

5081 

5311 

5542 

6772 

6002 

6232 

230 

9 

6462 

6692 

692) 

7151 

7380 

7609 

7838 

8067 

8296 

8525 

229 

190 

278754 

278982 

27921  1 

279439 

279667 

279895 

280123 

280351 

280578 

280806 

228 

I 

281033|281261 

231438281715 

281942 

282169 

2396 

2622 

2849 

3075 

227 

2 

3301!  3527 

3753   3979 

4205 

4431 

4656 

4882 

5107 

5332 

226 

3 

65571  5782 

6007 

6232 

6456 

6681 

6905 

7130 

7354 

7578 

225 

4 

7802  8026 

3249 

8473 

8696 

8920 

9143 

9366 

9589 

9812 

223 

6290035:290257 
8   2256   2478 

290180  j  2907*02 
2699  2920 

290925 
3141 

291147 
3363 

291369 
3584 

291591 
3804 

291813 
4025 

292034 
4246 

222 
221 

7 

4466   4637 

4907   5127 

5347 

6567 

6787 

6007 

6226 

6446 

220 

8 

6665 

68.34 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

9 

8853 

9071 

9289 

9507 

9725 

9943 

300161 

300378 

300595 

300813 

218 

Vtt 

301030 

301247 

301464 

301681 

301898 

3021  14 

302331 

302547 

302764 

302980 

217 

1 

31% 

3412 

3623 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

216 

2 

5351 

5566 

5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

215 

3 

7496 

7710 

7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

213 

4 

9630 

9343 

310056 

310263 

3L043I 

310693 

310906 

311118 

311330 

311542 

212 

6 

311754 

311966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

211 

g 

3867 

4078 

4239 

4499 

4710 

4920 

5130 

6340 

6551 

5760 

210 

7 

5970 

6130 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

209 

8 

8063 

8272 

8481   8639 

8898 

9106 

9314 

9522 

9730 

9938 

208 

9 

320146 

320354 

320562J320769 

320977 

321184 

321391  321598 

321805 

322012 

207 

210322219 

322426 

322633322839 

323046 

323252 

323458  323665 

323871 

324077 

206 

I 

4282  4438 

46941  4899 

5105 

5310 

5516 

5721 

5926 

6131 

205 

2 

6336  6541 

6745  6950 

7155 

7359 

7563 

7767 

7972 

8176 

204 

3 

3330  8583 

8787  i  8991 

9194 

9398 

9601 

9805 

330008 

330211 

203 

4 

330414  330617 

330819 

33  102-2  33  1-22." 

331427 

331630 

331832 

2034 

2236 

202 

5 

2438!  2640 

2842 

3044   3-246 

3447 

3649 

3850 

4051 

425? 

202 

6 

4454   4655 

5057 

5-257 

5458 

5653 

5859 

6059 

6260 

201 

7 

6460   666( 

686C 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

200 

8 

8456!  8656 

8855   9054   9'253 

9451 

9650 

9349 

340047 

34<r246 

199 

9 

340444  310642 

340841 

341039341237 

341435 

341632 

341830 

2028 

2225 

198 

•to 

0 

1 

a 

3 

4 

5 

6 

7 

8 

9 

DM. 

TABLE    I.      LOGARITHMS    OF    NUMBERS. 


Hal  O 

2 

3    4 

5 

6 

7  I  8 

0 

Diff. 

So 

342423 

342620 

3423  1  7 

343014  343212 

343409 

343606 

343802  '343999 

344196 

197 

i 

43'J2 

4589 

4785 

4981   6178 

5374 

5570 

6766J  6962 

6157 

1% 

2 

6353 

6549 

6744 

6939   7135 

7330 

7525 

7720  7915 

8110 

195 

3 

831)5 

8500 

8694 

8889  9083 

927£ 

9472 

9666  9860 

350054 

194 

4 

350248 

350442 

350636 

350829  351023 

351216 

351410 

351603!351796 

1989 

193 

5 

2183 

2375 

2568 

2761,  2954 

3147 

3339 

3532  '  3724 

3916 

193 

6 

4108 

4301 

4493 

4685 

4876 

6068 

5260 

5452,  5643 

5834 

192 

7 

6026 

6217 

64(>8 

6599 

6790 

6981 

7172 

7363   7554 

7744 

191 

8 

7935 

8125 

8316 

ssoe 

8696 

8886 

9076 

9266;  9456 

9646 

190 

9 

9835 

360025 

360215 

360404360593 

360783 

360972 

361161 

361350 

361539 

189 

230 

361728 

361917 

362105 

362294362482 

363671 

362859 

36304P 

3*53236 

363424 

188 

1 

3612 

3800 

3938 

4176;  4363 

4551 

4739 

4926   5113 

6301 

188 

2 

6488 

5675 

5862 

6049  6236 

6423 

6&0 

6796 

6983 

7169 

187 

3 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

186 

4 

9216 

9401 

9587 

9772 

9958 

370143 

370328 

370513 

370698 

370883 

186 

6 

371068 

371253 

371437 

371622371806 

1991 

2175 

2360 

2544 

2728 

184 

6 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184 

7 

4748 

4932 

5115 

5298 

6481 

6664 

5846 

6029 

6212 

6394 

183 

8 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

6216 

182 

9 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

380030 

181 

240 

380211 

380392 

380573 

380754 

380934 

381115 

381296 

381476 

381656 

381837 

181 

1 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

180 

2 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

6070 

6249 

5428 

179 

3 

5606 

5785 

5964 

6142 

6321 

6499 

6677 

6856 

7034 

7212 

178 

4 

7390 

7568 

7746 

7923 

8101 

8279 

8456 

8634 

8811 

8989 

178 

6 

9166 

9343 

9520 

9693 

9875 

390051 

39022.5 

390405 

390582 

390759 

177 

6 

390935 

391112 

391288 

391464 

391641 

1817 

1993 

2169 

2345!  2521 

176 

7 

2697 

2873 

3048 

3224 

3400 

3575 

3751 

3926 

4101 

4277 

176 

6 

4452 

4627 

4802 

4977 

6152 

5326 

6501 

6676 

6850 

6025 

176 

9 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

260 

397940 

398114 

398287 

398461 

398634 

398808 

398981 

399154 

399328 

399501 

173 

1 

9674 

9847 

400020 

400192 

400365 

400538 

400711 

400883 

401056 

401228 

173 

2 

401401 

401573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

172 

3 

3121 

3292 

3464 

3635 

3807 

3978 

4149 

4320 

4492 

4663 

171 

4 

4834 

5005 

6176 

6346 

6517 

6688 

5858 

6029 

6199 

6370 

171 

6 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

170 

6 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

169 

7 

9933 

410102 

410271 

410440 

410609 

410777 

410946 

411114 

411283 

411451 

169 

8 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

168 

9 

3300 

3467 

3635 

3S03 

3970 

4137 

4305 

4472 

4639 

4806 

167 

200 

414973 

415140 

415307 

415474 

415641 

415808 

415974 

416141 

416308 

4i6474 

167 

1 

6641 

6807 

6973 

7139 

7306 

7472 

7638 

7804 

7970 

8135 

166 

2 

8301 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

165 

3 

9956 

420121 

420286 

420451 

420616 

420781 

420945 

421110 

421275 

421439 

165 

4 

421604 

1768 

1933 

2097 

2261 

2426 

2590 

2754!  2918 

3082 

164 

5 

3246   3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

164 

6 

4882 

5045   5?08 

5371 

5534 

5697 

6860 

6023 

6186 

6349 

163 

7 

6511 

6674 

683o   6999 

7161 

7324 

7486 

7648 

7811 

7973 

162 

8 

8135 

8297 

8459 

86al 

8783 

8944 

9106 

9268 

9429 

9591 

162' 

9 

9752 

9914 

430075 

430236 

430398 

430559 

43072C 

430881 

431042 

431203 

161 

'^70 

431364 

431525 

431685 

431846 

432007 

•132167 

432328 

432488 

432649 

432809 

161 

1 

2969 

3130 

3290 

3450 

3610 

3770 

3930 

4090 

42494 

4409 

160 

2 

4569 

4729 

4888 

5048 

5207 

5367 

5526 

5685 

6845 

6004 

159 

3 

6163 

6322 

6481 

6640 

6799 

6957 

7116 

7275 

7433 

7592 

159 

4 

7751 

7909 

8067 

8226 

8384 

8542 

8701 

8859 

9017 

9175 

158 

5 

9333 

9491 

9648 

9806 

9964 

440122 

440279 

440437 

440594 

440752 

158 

6 

440909 

441066 

441224 

441381 

441538 

1695 

1852 

2009 

2166 

2323 

157 

7 

2480 

2637 

2793 

2950 

3106 

3263 

3419 

3576 

3732 

3889 

157 

8 

4045 

4201 

4357 

4513 

4669 

4825 

4981 

6137 

6293 

6449 

156 

9 

5604 

5760 

5915 

6071 

6226 

6382 

6537 

6692 

6848 

7003 

165 

No. 

0 

1 

2 

3 

A 

5 

6 

T 

8 

9 

Dltf. 

TABLE    I.      LOGARITHMS    OF    NUMBERS. 


215 


Ho 

O 

1 

3  • 

3 

4 

5 

6 

7 

8 

9 

Diff. 

280 

447158 

447313 

147468 

447623 

447778 

47933 

4-nss 

48242 

448397 

48552 

155 

1 

8706 

8861 

9015   9170   9324 

9478 

9633 

9787 

9941 

50095 

154 

2 

450219 

4504(13 

50557 

45071  1  450SG5 

51018 

51172 

51326 

451479 

1633 

154 

3 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3012 

3165 

153 

4 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

153 

5 

4845 

4997 

5150 

5302 

5454 

5606 

5758 

6910 

6062 

6214 

152 

6 

6366 

6518 

6670 

6821 

6973 

7125 

7276 

7428 

7579 

7731 

152 

7 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8940 

9091 

9242 

151 

8 

9392 

9543 

9694 

9845 

9995 

60146 

60296 

60447 

60597 

60748 

151 

9 

460898 

461048 

61198 

461348 

461499 

1649 

1799 

1948 

2098 

2248 

150 

290 

462398 

462548 

62697 

462847 

162997 

63146 

63296 

63445 

463594 

463744 

150 

j 

3893 

4042 

4191  1  4340 

4490 

4639 

4788 

4936 

5085 

5234 

149 

2 

6383 

5532 

5680 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

3 

6868 

7016 

7164 

7312 

7460 

7608 

7756 

7904 

8052 

8200 

148 

4 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9527 

9675 

148 

5 

9S22 

9969 

70116 

470263 

470410 

70557 

470704 

70851 

470998 

471145 

147 

6 

471292 

471438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

146 

7 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

8 

4216 

4362 

4508 

4653 

4799 

4944 

5090 

5235 

5381 

5526 

146 

9 

6671 

5816 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

145 

300 

477121 

477266 

77411 

477555 

477700 

77844 

477989 

478133 

478278 

478422 

145 

1 

8566 

8711 

8855 

8999 

9143 

9287 

9431 

9575 

9719 

9863 

144 

2 

480007 

480151 

80294 

480438 

480582 

80725 

480869 

481012 

481  156 

481299 

144 

3   1443 

1586 

1729 

1872 

2016 

2159 

2302 

2445 

2588 

2731 

143 

4 

2874 

3016 

3159 

3302 

3445 

3587 

3730 

3872 

4015 

4157 

143 

5 

4300 

4442 

4585 

4727 

4869 

6011 

5153 

5295 

6437 

6579 

142 

6 

5721 

5863 

6005 

6147 

6289 

6430 

6572 

6714 

6855 

6997 

142 

7 

7138 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

8269 

8410 

141 

8 

8551 

8692 

8833 

8974 

9114 

9255 

9396 

9537 

9677 

9818 

141 

9 

9958 

490099 

490239 

490380 

490520 

490661 

490801 

490941 

491081 

491222 

140 

310 

491362 

491502 

491642 

491782 

491922 

492062 

492201 

492341 

492481 

492621 

140 

1 

2760 

2900 

3040 

3179 

3319 

3458 

3597 

3737 

3876 

4015 

139 

2 

4155 

4294 

4433 

4572 

4711 

4850 

4989 

5128 

5267 

5406 

139 

3 

5544 

5683 

6822 

5960 

6099 

6238 

6376 

6515 

6653 

6791 

139 

4 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

138 

6 

8311 

8448 

8586 

8724 

8862 

8999 

9137 

9275 

9412 

9550 

138 

6 

9687 

9824 

9962 

500099 

500236 

500374 

500511 

500648 

500785 

500922 

137 

7 

601059 

501196 

501333 

1470 

1607 

1744 

1880 

2017 

2154 

2291 

137 

8 

2427 

2564 

2700 

2837 

2973 

3109 

3246 

3382 

3518 

3655 

136 

9 

3791 

3927 

4063 

4199 

4335 

4471 

4607 

4743 

4678 

6014 

136 

320 

605150 

505286 

505421 

505557 

505693 

505828 

505964 

506099 

506234 

506370 

136 

1 

6505 

6640 

6776 

6911 

7046 

7181 

7316 

7451 

7586 

7721 

135 

2 

7856 

7991 

8126 

8260 

8395 

8530 

8664 

8799 

8934 

9068 

135 

3 

9203 

9337 

9471 

9606 

9740 

9874 

510009 

510143 

510277 

510411 

134 

4 

510545 

510679 

510813 

510947 

511081 

511215 

1349 

1482 

1616 

1750 

134 

5 

1883 

2017 

2151 

2284 

2418 

2551 

2684 

2818 

2951 

3084 

133 

6 

3218 

3351 

3484 

3617 

3750 

3883 

4016 

4149 

4282 

4415 

133 

7 

4548 

46S1 

4813 

4946 

5079 

5211 

5344 

5476 

5609 

5741 

133 

e 

5874 

6006 

6139 

6271 

6403 

6535 

6668 

6800 

6932 

7064 

132 

9 

7196 

7328 

7460 

7592 

7724 

7855 

7987 

8119 

8251 

8382 

132 

330 

&  8514 

518646 

518777 

518909 

51904( 

519171 

519303 

519434 

519566 

519697 

131 

11  9828!  9959 

520090 

520221,520353 

520484 

520615 

520745 

520876 

621007 

131 

2621138 

521269 

1400 

1530 

166 

1792 

1922 

2053 

2183 

2314 

131 

a 

2444 

2575 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

3616 

130 

4 

3746 

3876 

4006 

4136 

4266 

4396 

4526 

4656 

4785 

4915 

130 

I 

5045 

5174 

5304 

5434 

5563 

5693 

5822 

6951 

608 

6210 

129 

e 

6339 

6469 

6598 

6727 

6856 

6985 

7114 

7243 

7372 

7501 

129 

7 

7630 

7759 

7888 

8016 

8145 

8274 

8402 

8531 

8660 

8788 

129 

1 

1  8917 

9045 

9174 

9302 

9430 

9559 

9687 

9815 

9943 

530072 

128 

£ 

>  53020C 

530328 

530456 

530584 

530712 

530840 

530968 

531096 

531223 

135 

128 

Na   0 

1 

J8 

3 

Wff. 

216 


TABLE    I.     LOGARITHMS    OF    NUMBERS, 


No. 

o 

1 

2 

Dlff. 

340 

531479 

531607 

631734 

531862  531990 

532117 

532245 

532372  532500  632627 

128 

1 

2754 

2882 

3009 

3136 

3264 

3391 

3518 

3646 

3772 

3&99    127 

2 

4026 

4153 

4280 

4407 

4534 

4661 

47*7 

4914 

5041 

5167    127 

3 

5294 

5421 

6547 

6674 

6800 

5927 

6053 

6IM3 

6306 

6432    (26 

4 

6558 

6685 

6811 

6937 

7063 

7189 

7315 

7441 

7567 

7693 

126 

6 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951!  126 

6 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

540079  '540204    126 

7 

540329 

54W55 

540580 

C40705 

640S30 

640956 

641080641206 

1330!     1454    125 

8 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

9 

2825 

2950 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

124 

3ft: 

544068 

644192 

644316 

644440 

644564 

M46Q6 

644812 

644936 

645060 

645183 

124 

1 

6307 

5431 

6555 

6678 

6802 

6925 

6049 

6172 

6296 

6419 

124 

2 

6543 

££££ 
DODO 

6789 

6913 

7036 

7159 

72.V2 

7405 

7529 

7652 

123 

3 

7775 

7898 

8021 

6144 

8267 

8389 

651-2 

S635 

87f8 

8881 

123 

4 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9S6I 

9984 

560106 

123 

6 

660228 

650351 

650473 

660595 

650717 

660840 

650962 

651084 

651206 

1328|  122 

6 

1450 

1572 

1694 

1816 

1938 

WOO 

2181 

2303 

2425 

2547    122 

7 

2668 

2790 

2911 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

121 

8 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

121 

9 

6094 

6215 

6336 

6457 

6678 

6699 

6820 

6940 

6061 

6182 

121 

360 

666303 

656423 

666544 

666664 

6667S5 

566905 

657026 

667146 

657267 

657387 

120 

7607 

7627 

7748 

76631     7988 

8108 

8228 

63491     6469 

8589    120 

2 

8709 

8S29 

8<M8 

9068      9168 

9308 

9428 

9548|     9667 

9787    120 

b 

9007 

660026 

660146 

660265  660385 

100604 

660C2-4 

660743  660^63660982  i  119 

4 

661101 

1221 

1340 

1459 

1578 

1606 

1617 

1936      2055      21741  119 

5 

2293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244      3362    110 

6 

3481 

3600 

3718 

3837 

3955 

4074 

4192 

4311 

4429      4548    119 

7 

4666 

4784 

4903 

6021 

6139 

6257 

6376 

6494 

6612 

6730    118 

8 

6848 

5966 

6084 

6202 

6320 

C437 

6555 

6673 

6791 

6909 

118 

9 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7649 

7967 

8064 

118 

370 

568202 

668319 

668436 

66S564 

568671 

568768 

666905 

569023 

669140 

569257 

117 

1 
2 

9374 
670543 

9491      96f*8|     97251     9842 
670660  67(»776  6708931671010 

9959,670076  670193  570309  670426'   117 
671126      1243      1359      1476      1592    117 

3 

1709 

1825 

1942 

2058 

2174 

2291      2407 

2:V23      2639      2756:  118 

4 

2872 

2988 

3104 

3220 

3336 

3452      3568 

3684      35,/u      3yi5    llfi 

6 

4031 

4147 

4263 

4379 

4494 

4610|     4726 

4841!     4957      6072    116 

6 

6188 

5303 

6419 

6534 

6650 

6765      6680 

6996      6111      6226    116 

7 

6341 

6457 

6572 

6687 

6802 

6917      7032 

7147      7202      7377;  116 

8 

7492 

7607 

7722 

7836 

7951 

8066 

8181      8295 

6410      6525!  116 

9 

6639 

8764 

8668 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

114 

380 

679784 

679898 

680012 

680126'580241 

680355  680469 

6805«3 

680697 

680811 

114 

1 

580925 

581039 

1153 

1267 

1381 

1495!     J6II8 

V£t\     IKifi      11*50)  |14 

2 

2063 

2177 

2291 

2404 

2518 

2631      2745 

368.     WJ7X      30*5    114 

3 

3199 

3312 

3426 

3539 

3652 

3765  i     as79 

3992     4106      4218      13 

4 

4331 

4444 

4557 

4670 

4763 

4S96J     6C09 

6122      G'/:«      6348^     J3 

6 

6461 

5574 

6686 

5799 

6912 

6<r24      6137 

6250!     6362     64751     13 

6 

6587 

6700 

6812 

6925 

7037 

7149 

7262      7374      74*6      7599 

.12 

7 

7711 

7823 

7935 

8047 

81  6( 

8272 

83-vli     8496      6608!     8720 

112 

8 

8832 

8944 

9056 

9167)     9279 

9391 

9503      9615!     9726  !     9*38 

112 

9 

9950 

690061 

690173 

690284690396 

590507 

590619  690730  690842  690953 

112 

390 

691  06r 

691176 

691287 

591399591510 

591621 

691732 

691843  691955  692066 

111 

1 

2177 

2288 

2399 

2510,     2621 

2732 

2843      2954      3064      3175J  111 

3286 

3397 

3508 

3618|     3729 

384(1 

3950      4061 

417        4282    111 

| 

4393 

4503 

4614 

4724      4834 

4945      6055!     5165 

5276      53*6    110 

4 

6496 

5606 

6717 

5827      6937 

6047]     6157      6267      6377      64-^7    110 

6 

6597 

6707 

6817 

6927      7037 

7146      7256      7366      7476      7r.,s6    110 

( 

7695 

7805 

7914 

8024      8134 

6243      8353      ^462      8572      86*1    110 

879 

890C 

9009 

9119      9228 

9337      9446      9556      9665      9774    109 

8 

9883 

9992 

600101 

600210600319 

600428  6tX»f>37  6006-46  600755  600-64    109 

9 

600973 

601  0& 

1191 

1299      1408 

1517 

1625      1734,     1843,     1961;  109 

No 

•          1 

9 

3456 

7     1     8          9 

Diff. 

TABLE    I,      LOGARITHMS     OF     NUMBERS. 


217 


No.|  0 

1 

3    3  |  4 

5  |  6 

7  |  § 

9 

Diff 

400602060 

602169 

602277  '602336  '6024  94 

602603 

602711 

602819602928 

603036 

108 

1   3144   3253 

3361   34691  3577 

3686 

3794 

3902 

4010 

4118 

108 

21  4'226   4334 

4442   4550 

4658 

4766 

4874 

4932 

5039 

5197 

108 

3>  53(15   5413 

5521  :  5628 

5736 

5844 

5951 

6059 

6166 

6274 

103 

4  6331;  6439 

6596  6704 

.6811 

6919 

7026 

7133 

7241 

7348 

107 

6!  7455   7562 

7669   7777 

7884 

7991 

8098 

8205 

8312 

8419 

107 

6  8526   SG33 

8740 

8847 

8954 

9061 

9167 

9274 

9331 

9488 

107 

7 

9594!  9701 

9808  9914 

610021 

610128 

610234 

610341 

610447 

610554 

107 

8 

610660:610767 

610.373  610979 

1036 

1192 

1298 

1405 

1511 

1617 

106 

9 

1723 

1829 

1936 

2042 

2148 

2254 

2360 

2466 

2572 

2678 

106 

410 

612784 

612390 

612996 

613102 

613207 

613313 

613419 

613525 

613630 

613736 

106 

1 

3842 

3947 

40531  4159 

4264 

4370 

4475 

4581 

4686 

4792 

106 

2 

4897 

5003 

5108 

5213 

5319 

5424 

5529 

5634 

5740 

5845 

105 

3 

5950 

6055 

6160 

6265 

6370 

6476 

6581 

6636 

6790 

6895 

105 

4 

7000 

7105 

7210 

7315 

7420 

7525 

7629 

7734 

7839 

7943 

105 

5 

8048 

8153 

8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

105 

6 

9093 

9198 

9302 

3406 

9511 

9615 

9719 

9824 

9928 

620032 

104 

7 

620136 

620240 

620344  620443 

620552 

620656 

620760 

620864 

620968 

1072 

104 

8 

11761  1280 

1384 

1488 

1592 

1695 

1799 

1903 

2007 

2110 

104 

9 

2214 

2318 

2421 

2525 

2623 

2732 

2835 

2939 

3042 

3146 

104 

420 

623249 

623353 

623456 

623559 

623663 

623766 

623869 

623973 

624076 

624179 

103 

1 

4232 

4335 

4438 

4591 

4695 

4798 

4901 

5004 

5107 

5210 

103 

2 

5312 

5415 

5518 

5621 

5724 

5827 

5929 

6032 

6135 

6238 

103 

3 

6340 

6443 

6516 

6643 

6751 

6853 

6956 

7058 

7161 

7263 

103 

4 

7366 

7463 

7571 

7673 

7775 

7878 

7980 

8082 

8185 

8287 

102 

5 

8339 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

9206 

9308 

102 

6 

9410 

9512 

9613  9715 

9817 

9919 

630021 

630123 

630224 

630326 

102 

7 

630423 

630530 

630631:630733 

630835 

630936 

1038 

1139 

1241 

1342 

102 

8 

1444 

1545 

1647   1743 

1849 

1951 

2052 

2153 

2255 

2356 

101 

9 

2457 

2559 

2660  2761 

2862 

2963 

3064 

3165 

3266 

3367 

101 

430 

63346S 

633569 

633670  633771 

633872 

633973 

634074 

634175 

634276 

634376 

101 

1 

4477 

4578 

4679 

4779 

4880 

4981 

5081 

5182 

5283 

5383 

101 

2 

5484 

5584 

5635 

5785 

5886 

5986 

6087 

6187 

6287 

6388 

100 

8  6483 

6533 

6638 

6789 

6889 

6989 

7089 

7189 

7290 

7390 

100 

4  7490 

7590 

7690 

7790 

7890 

7990 

8090 

8190 

8290 

8389 

100 

5  8439 

8539 

6639 

8789 

8888 

8988 

9038 

9188 

9287 

9387 

100 

6|  9436 

9586 

9686 

9785 

9885 

9984 

640084 

640183 

640283 

640382 

99 

7  1  640431 

640581 

640680 

640779 

640379 

640978 

1077 

1177 

1276 

1375 

99 

8   1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 

99 

9  2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

3255 

3354 

99 

440643453 

643551 

643650 

643749 

643347 

643946 

644044 

644143 

644242 

644340 

98 

I   4439 

4537 

4636 

4734 

4832 

4931 

5029 

5127 

5226 

5324 

98 

2  5422 

5521 

5619 

5717 

5815 

5913 

6011 

6110 

6208 

6306 

98 

3  6404 

6502 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

98 

41  7333 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

8165? 

8262 

98 

6 

8360 

8458 

8555 

8653 

8750 

8848 

8945 

9043 

9140 

9237 

97 

6 

9335 

9432 

9530 

9627 

9724 

9821 

9919 

650016 

650113 

650210 

97 

7 

650303 

650405 

650r>02 

630599 

650696 

650793 

650890 

0987 

1084 

1181 

97 

8 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

1956 

2053 

2150 

97 

9 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3019 

3116 

97 

460 

653213 

653309 

653405 

653502 

653598 

653695 

653791 

653888 

653984 

654080 

96 

1 

4177 

4273 

4369  44651  4562 

4658 

4754 

4850 

4946 

5042 

96 

2 

5133 

5235 

5331:  5427   5523 

5619 

5715 

5810 

5906 

6002 

96 

3 

6093|  6194 

6290  1  6336 

6482 

6577 

6673 

6769 

6864 

6960 

96 

4 

7056 

7152 

72471  7343 

7438 

7534 

7629 

7725 

7820 

7916 

96 

5 

8011 

8107 

8202 

8293 

8393 

8488 

8584 

8679 

8774 

8870 

95 

6 

8965 

9060 

9155 

9250 

9346 

9441 

9536 

9631 

9726 

9821 

95 

7 

9916 

66001  1 

660106 

660201 

660296 

660391 

660486 

660581 

660676 

660771 

95 

8 

660865 

0960 

1055 

1150 

1245   1339 

1434 

1529 

1623 

1718 

95 

_9 

1813 

1907 

2002 

2096 

2191  j  2236 

2380 

2475 

2569 

2663 

95 

•a. 

~O 

1 

a 

3 

Dlff 

218 


TABLE    1.      LOGARITHMS    OF    NUMBERS. 


No. 

0 

1 

3 

9 

IAS. 

460 

562758 

562852 

662947 

63041 

G63135 

663230 

663324:663418  663512 

663607 

94 

1 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

4360 

4454 

4548 

94 

2 

4642 

4736 

4830 

4924 

5018 

5112 

5206 

5299 

5393 

5487 

94 

3 

5581 

5675 

6769 

6862 

5956 

6050 

6143 

6237 

6331 

6424 

94 

4 

6518 

6612 

6705 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

M 

5 

7453 

7546 

7640 

7733 

7826 

7920 

8013 

8106 

8199 

8293 

93 

6 

8386 

8479 

8572 

8665 

8759 

8852 

8945 

9038 

9131 

9224 

93 

7 

9317   9410 

9503 

9f96 

9689 

9782 

9875 

9967 

670060 

670153 

93 

8 

570246 

70339 

670431 

670524 

670617 

670710 

670802 

670895 

0988 

1080 

93 

9 

1173 

1265 

1358 

1451 

1543 

1636 

1728 

1821 

1913 

2005 

93 

470 

672098 

672190 

672283 

672375 

672467 

672560 

672652 

672744 

672836 

672929 

92 

1 

3021 

3113 

3205 

3297 

3390 

3482 

3574 

3666 

3758 

3850 

92 

2 

3942 

4034 

4126 

4218 

4310 

4402 

4494 

4586 

4677 

4769 

92 

3 

4861 

4953 

5045 

6137 

.5228 

5320 

5412 

6503 

5595 

'5687 

92 

4 

5778 

5870 

5962 

6053 

6145 

6236 

6328 

6419 

6511 

6602 

92 

5 

6694 

6785 

6876 

6968 

7059 

7151 

7242 

7333 

7424 

7516 

91 

6 

7607 

7698 

7789 

7881 

7972 

8063 

8154 

8245 

8336 

8427 

91 

7 

8518 

8609 

8700 

8791 

8882 

8973 

9064 

9155 

9246 

9337 

91 

8 

9428 

9519 

9610 

9700 

9791 

9882 

9973 

680063 

680154 

680245 

91 

9 

680336 

680426 

680517 

680607 

680698 

680789 

680879 

0970 

1060 

1151 

91 

480 

681241 

681332 

681422 

681513 

681603 

681693 

681784 

681874 

681964 

682055 

90 

1 

2145 

2235 

2326 

2416 

2506 

2596 

2686 

2777 

2867 

2957 

90 

2 

3047 

3137 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

3 

3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

90 

4 

4845 

4935 

5026 

6114 

5204 

5294 

6383 

6473 

6563 

5652 

90 

6 

5742 

6831 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

89 

6 

6636 

6726 

6815 

6904 

6994 

7083 

7172 

7261 

7351 

7440 

89 

7 

7529 

7618 

7707 

7796 

7886 

7975 

8064 

8153 

8242 

8331 

89 

8 

8420 

8509 

8598 

8687 

8776 

8865 

8953 

9042  9131 

9220 

89 

9 

9309 

9398 

9486 

9576 

9664 

9753 

9841 

99301690019 

690107 

89 

490 

890196 

690285 

690373 

690462 

690550 

690639 

690728 

690816 

690905 

690993 

69 

I 

1081 

1170 

1258 

1347 

1435 

1524 

1612 

1700 

1789 

1877 

68 

2 

1965 

2053 

2142 

2230 

2318 

2406 

2494 

2583 

2671 

2759 

88 

3 

2847 

2935 

3023 

3111 

3199 

3287 

3375 

3463 

3551 

3639 

88 

4 

3727 

3815 

3903 

3991 

4078 

4166 

4254 

4342 

4430 

4517 

88 

6 

4605 

4693 

4781 

4868 

4956 

5044 

5131 

5219 

5307 

6394 

88 

6 

5482 

5569 

5657 

5744 

5832 

6919 

6007 

6094 

6182 

6269 

87 

7 

6356 

6444 

6531 

6618 

6706 

6793 

6880 

6968 

7055 

7142 

87 

8 

7229 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

87 

9 

8101 

8188 

8275 

8362 

8449 

8536 

8622 

8709 

8796 

8883 

87 

500 

698970 

699057 

699144 

699231 

699317 

699404 

699491 

699578 

699664 

699751 

87 

1 

9838 

9924 

700011 

700098  700184 

700271 

700358 

700444 

700531 

700617 

87 

2 

700704 

700790 

0877 

0963 

1050 

1130 

1222 

1309 

1395 

1482 

86 

j 

1568 

1654 

i741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

86 

4 

2431 

2517 

2603 

2689 

2775 

2861 

2947  3033  3119 

3205 

86 

5 

3291 

3377 

3463 

3549 

3635 

3721 

3807 

3893 

3979 

4065 

86 

6 

4151 

4236 

4322 

4408 

4494 

4579 

4665 

4751 

4837 

4922 

86 

7 

5008 

5094 

6179 

5265 

6350 

5436 

6522 

5607 

5693 

6778 

86 

8 

5864 

5949 

6035 

6120 

6206 

6291 

6376 

6462 

6547 

6632 

85 

9 

6718 

6803 

6888 

6974 

7059 

7144 

7229 

7315 

7400 

7485 

85 

510 

707570 

707655 

707740 

707826 

707911 

707996 

708081 

708166 

708251 

708336 

85 

] 

8421 

8506 

8591 

8676  8761 

8846 

8931 

9015 

9100 

9186 

85 

,< 

9270 

9355 

9440 

9524  9609 

9694 

9779 

9863 

9948 

710033 

85 

| 

710117 

710202 

710287 

710371  710456 

710540 

710625 

710710 

710794 

0879 

85 

I 

0963 

1048 

1132 

1217   1301 

1385 

1470 

1554 

1639 

1723 

84 

5 

1807 

1892 

1976 

2060  2144 

2229 

2313 

2397 

2481 

2566 

S4 

6 

2650 

2734 

2818 

2902  2986 

3070 

3154 

3238 

3323 

3407 

84 

j 

3491 

3575 

3659 

3742  3826 

3910 

3994 

4078 

4162 

4246 

84 

8 

4330 

4414 

4497 

4581   4665 

4749 

4833 

4916 

5000 

5084 

84 

9 

5167 

5251 

53351  5418  5502 

5536 

5669 

5753 

5836 

5920 

84 

Ho 

0 

1 

334 

0 

6 

7 

6 

9 

JMff. 

TABLE    I.      LOGARITHMS    OF    NUMBERS. 


219 


No.l  0 

1 

3  i 

3 

4 

,5 

6 

7 

8 

9 

Difl. 

620  716003 

16087 

716170 

716254 

716337 

16421 

16504 

16588 

716671 

716754 

~K 

1 

6838 

6921 

7004 

7088 

7171 

7254 

7338 

7421 

7504 

7587 

83 

2 

7671 

7754 

7837 

7920 

8003 

8086 

8169 

8253 

8336 

8419 

83 

3 

8502 

8585 

8668 

8751 

8834 

8917 

9000 

9083 

9165 

9248 

83 

4 

9331 

9414 

9497 

9580 

9663 

9745 

9328 

9911 

9994 

720077 

83 

5 

720159 

720242 

720325 

720407 

720490 

20573 

20655 

720733 

720321 

0903 

83 

6 

0986 

1068 

1151 

1233 

1316 

1398 

1481 

1563 

1646 

1728 

82 

7 

1811 

1893 

1975 

2058 

2140 

2222 

2305 

2387 

2469 

2552 

82 

8 

2634 

2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

82 

9 

3456 

3533 

3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

82 

630 

724276 

724358 

724440 

724522 

724604 

24685 

724767 

24849 

~24931 

725013 

82 

1 

5095 

5176 

5258 

5340 

5422 

5503 

5585 

5667 

5748 

5830 

82 

2 

5912 

5993 

6075 

6156 

6233 

6320 

6401 

6483 

6564 

6646 

82 

3 

6727 

6809 

6890 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

81 

4 

7541 

7623 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

81 

6 

8354 

8435 

8516 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

81 

6 

9165 

9246 

9327 

9408 

9489 

9570 

9651 

9732 

9813 

9893 

81 

7 

9974 

730055 

730136 

730217 

730298 

730378 

730459 

~30540 

730621 

730702 

81 

8 

730782 

0363 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

81 

9 

1589 

1669 

1750 

1830 

1911 

1991 

2072 

2152 

2233 

2313 

81 

640 

732394 

732474 

732555 

732635 

732715 

732796 

732876 

732956 

733037 

733117 

80 

1 

3197 

3278 

3358  3438 

3518 

3598 

3679 

3759 

3839 

3919 

80 

2 

3999 

4079 

4160 

4240 

4320 

4400 

4480 

4560 

4640 

4720 

80 

3 

4800 

4880 

4960 

5040 

6120 

5200 

5279 

6359 

5439 

5519 

80 

4 

5599 

5679 

6759 

5838 

5918 

5998 

6078 

6157 

6237 

6317 

80 

6 

6397 

6476 

6556 

6635 

6715 

6795 

6874 

6954 

7034 

7113 

80 

6 

7193 

7272 

7352 

7431 

7511 

7590 

7670 

7749 

7829 

7908 

79 

7 

7987 

8067 

8146 

8225 

8305 

8384 

8463 

8543 

8622 

8701 

79 

8 

8781 

8860 

8939 

9018 

9097 

9177 

9256 

9335 

9414 

9493 

79 

9 

9572 

9651 

9731 

9310 

9889 

9968 

740047 

740126 

740205 

740284 

79 

550 

740363 

740442 

710521 

740600 

740678 

740757 

740836 

740915 

740994 

741073 

79 

1 

1152 

1230 

1309 

1388 

1467 

1546 

1624 

1703 

1782 

1860 

79 

2 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2647 

79 

3 

2725 

2804'  2332 

2961 

3039 

3118 

3196 

3275 

3353 

3431 

78 

4 

3510 

3533  3667 

3745 

3823 

3902 

3930 

4058 

4136 

4215 

78 

6 

4293 

4371-  4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

78 

6 

5075 

6153 

5231 

5309 

5337 

5465 

5543 

5621 

5699 

5777 

78 

7 

5855 

5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

78 

8 

6634 

6712 

'6790 

6363 

6945 

7023 

7101 

7179 

7256 

7334 

78 

9 

7412 

7489 

7567 

7645 

7722 

7800 

7878 

7955 

8033 

8110 

78 

r>60  748183 

748266 

748343 

748421 

748498 

748576 

748653 

748731 

748808 

748885 

77 

11  8963 

9<I4( 

9118 

9195 

9272 

9350 

9427 

9504 

9582 

9659 

77 

2  9736 

9814 

9391 

9968 

750045 

750123 

750200 

750277 

750354 

750431 

77 

3'75050S 

750586 

750663 

750740 

0817 

0894 

0971 

1048 

1125 

1202 

77 

4 

1279 

1356 

1433 

1510 

1587 

1664 

1741 

1818 

1895 

1972 

77 

5 

2043 

2125 

2202 

2279 

2356 

2433 

2509 

2586 

2663 

2740 

77 

6 

2316 

2393 

2970 

3047 

3123 

3200 

3277 

3353 

3430 

3506 

77 

7 

3533 

3660 

3736 

3813 

3889 

3966 

4042 

4119 

4i95 

4272 

77 

8 

4343 

4425 

4501 

4578 

4654 

4730 

4807 

4883 

4960 

5036 

76 

9 

5112 

518S 

5265 

5341 

5417 

6494 

5570 

5646 

6722 

5799 

76 

570 

75587 

755951 

756027 

756103 

756180 

756256 

756332 

756408 

756484 

756560 

76 

1 

6636J  6712 

6788 

6364 

6940 

7016 

7092 

7168 

7244 

732f 

76 

2 
3 

7396  7475 
8155  823C 

7548 
8306 

7624 

8382 

7700 
8458 

7775 
8533 

7851 
8609 

7927 
8685 

8003 
876 

8079 
8836 

76 
76 

4 

8912  89.3S 

9063 

9139  9214 

9290 

9366 

944 

9517 

9592 

76 

6 

9G63   974c 

9819 

9394   997( 

76004 

760121 

760196 

760272 

760347 

75 

6 

76M22  76049,; 

760573:  760649  i  760724 

079 

0875 

0950 

1025 

110 

75 

7 

1176   1251 

1326 

140* 

1477 

155 

1627 

1702 

1778 

1853 

75 

8 

192c 

200C 

207fc 

215C 

222b 

230 

2378 

2453 

2529 

2604 

75 

I 

267S 

'2751 

282S 

2904 

2978 

305 

3128 

32iG 

3278 

3353 

75 

No 

0 

1 

a 

3 

4 

5 

6 

7 

8 

9 

Dtff. 

220 


TABLE 


LOGARITHMS     OF    NUMBERS. 


No. 

0 

1 

53 

3    4 

5 

6 

7 

8 

9 

Dlff.|j 

1586 

63428 

"63503 

"63578 

763653  ;  763727 

763802 

763877 

763952 

764027 

764101 

-75 

I  1 

4176 

425! 

4326 

4400   4475 

4550 

4624 

4699 

4774 

4M8 

76 

2 

4923 

4993 

5072 

5147 

6221 

5296 

5370 

5445 

5520 

5594 

75 

3 

5669 

5743 

5ttl8 

5892 

5U66 

6(»41 

6115 

6190 

6264 

6338 

74 

4 

5413 

6487 

6562 

6636 

6710 

6785 

6659 

6933 

7007 

7082 

74 

6 

7156 

7230 

73(14 

7379 

7453 

7527 

7601 

7675 

7749 

7823 

74 

6 

7895 

7972 

8046 

8120 

8194 

6268 

8342 

8416 

8490 

8564 

74 

7 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

9156 

9230 

9303 

74 

8 

9377 

9451 

9525 

9599 

9673 

9746 

9820 

9894 

9968 

770042 

74 

9 

770115 

770189 

770263 

770336 

770410 

770484 

770557 

770631 

770705 

0778 

74 

590 

770852 

770926 

770999 

771073 

771146 

771220 

771293 

771367 

771440 

771514 

74 

1 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

73 

3 

2322 

2395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

73 

3 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

73 

4 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

6 

4517 

4590 

4663 

4736 

4809 

4882 

4955 

5028 

5100 

5173 

73 

6 

5246 

6319 

6392 

5465 

553S 

5610 

5683 

5756 

5829 

6902 

73 

7 

5974 

6047 

6120 

6193 

6265 

6333 

6411 

6483 

6556 

6629 

73 

8 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

73 

9 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

6079 

72 

600 

778151 

778224 

778296 

778368 

778441 

778513 

778585 

778658 

778730 

778802 

72 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

72 

2 

9596 

9669 

9741 

9813 

9885 

9957 

780029 

780101 

780173 

780245 

72  1 

3 

780317 

780389 

780461 

780533  7&)605 

780677 

0749 

0821 

0893 

0965 

72 

4 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

72 

5 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

72 

6 

2473 

2544 

2616 

2688 

2759 

2831 

2902 

2974 

3046 

3117 

72 

7 

3189 

3260 

3332 

3403 

3475 

3546 

3618 

3889 

3761 

3832 

71 

8 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

71 

9 

4617 

4689 

4760 

48?  1 

490S 

4974 

5045 

5116 

5187 

5259 

71 

610 

785330 

785401 

785472 

785543 

785615 

785686 

785757 

785828 

785899 

785970 

71 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6538 

6609 

6680 

71 

2 

6751 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

7319 

7390 

71 

3 

7460 

7531 

7602 

7673 

7744 

7815 

7885 

7956 

8027 

8098 

71 

4 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

6734 

8804 

71 

5 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

71 

e 

9581 

9651 

9722 

9792 

9863 

9933 

790004 

790074 

790144 

790215 

70 

790285 

790356 

790426 

790496  790567 

790637 

0707 

0778 

0848 

0918 

70 

8 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1550 

1620 

70 

£ 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

70 

620 

792392 

792462 

792532 

792602 

792672 

792742 

792812 

792SS2 

792952 

793022 

70 

3092 

3162 

3231 

330f|  3371 

3441 

3511 

3581 

3651 

3721 

70 

i 

3790 

3860 

3930 

4000  4070 

4139 

4209 

4279 

4349 

4418 

70 

i 

4488 

4558 

4627 

4697  4767 

4836 

4906 

4976 

5045 

5115 

70 

t 

5185 

5254 

5324 

5393!  5463 

5532 

5602 

5672 

5741 

5811 

70 

i 

5880 

5949 

6019 

6088!  6158 

6227 

6297 

6366 

6436 

6505 

69 

6 

6574 

6644 

6713 

6782!  6852 

6921 

6990 

7060 

7129 

7198 

69 

j 

7268 

7337 

7406 

74751  7545 

7614 

7683 

7752 

7821 

7890 

69 

8 

7960 

8029 

8098 

8167 

8236 

8305 

8374 

8443 

8513 

8582 

69. 

9 

8651 

8720 

8789 

8858 

8927 

6996 

9065 

9134 

9203 

9272 

69 

630 

799341 

799409 

799478 

799547 

799616 

799685 

799754 

799823 

799892 

799961 

69 

800029 

800098 

800  1G7 

800236  80030f 

800373 

800442  80051  1 

800580 

800648 

69 

• 

0717 

0786 

0854 

0923  0992 

1061 

1129 

1198 

1268 

1335 

69 

j 

1404 

1472 

1541 

1609   1678 

1747 

1815 

1884 

1952 

2021 

69 

^ 

2089 

2158 

2226 

2295  2363 

2432 

2500 

2568 

2637 

2705 

68 

2774 

2842 

2910 

2979   3047 

3116 

3184 

3252 

3321 

3389 

68 

I 

3457 

3525 

3594 

3662  3730 

3798 

3%7 

3935 

4003 

4071 

68 

1 

4139 

4208 

4276 

4344  4412 

44.90 

4548 

4616 

4685 

4753 

68 

1 

4821 

4889 

4957 

5025  5093 

5161 

5229 

5297 

5365 

5433 

68 

I 

5501 

5569 

5637 

5705  5773 

5841 

5908 

5976 

6044 

6112 

68 

Na 

0 

1 

a 

3    4 

5 

6 

7 

6 

9 

Dlff. 

TABLE 


LOGARITHMS     OF     NUMBERS. 


221 


No.!  O  !  1 

a  I 

3 

9 

Diff. 

!  640  806  1801  8062  1  8 

306316 

30f>:^4 

306451 

806519 

806587 

06655 

306723 

806790 

68 

1   6858 

69  6 

6994 

7061 

7129 

7197 

7264 

7332 

7400 

7467 

68 

2  7535 

7603 

7670 

7738 

7806 

7873 

794; 

8008 

8076 

8143 

68 

3  8211 

8279 

8346 

8414 

8481 

8549 

861  ; 

8684 

8751 

8818 

67 

4  8886 

8953 

9021   9088 

9156 

9223 

9290 

9358 

9425 

9492 

67 

5'  9560 

9827 

9694   97621  9829 

9896 

9964 

810031 

810098 

810165 

67 

6;810233;810300 

10367  810434 

810501 

810569 

810636 

0703 

0770 

0837 

67 

7 

0904   0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

67 

.  8 

1575   1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

67 

9 

2245 

2312 

2379 

2145 

2512 

2579 

2646 

2713 

2780 

2847 

67 

650 

812913 

812980 

313047 

813114 

813181 

813247 

813314 

813381 

13448 

813514 

67 

1 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

67 

2 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

67 

3 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

5378 

5445 

5511 

66 

4 

5578 

5(544 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

66 

5 

6241 

6:308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6838 

66 

6  6904 

6970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

66 

71  7565 

7631 

7698 

'  7764 

7830 

7896 

7962 

8028 

8094 

8160 

66 

8  8226 

82'  12 

8358 

84241  8490 

85-56 

8622 

8638 

8754 

8820 

66 

9  8885 

8951 

9017 

9083 

9149 

92i5 

9281 

9346 

9412 

9478 

66 

660'  8  19544 

819610 

819676819741 

819807 

819873 

819939 

820004 

820070 

820136 

66 

1  8202(11 
2  0858 

820267 
0924 

820333820399 
0939'  1055 

820464 
1120 

320530 
1186 

820595 
1251 

0661 
1317 

0727 
1382 

0792 
1448 

66 
66 

3 

1514 

1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

65 

4 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

2626 

2691 

2756 

65 

5 

2822 

28S7 

2952 

3018 

3((83 

3148 

3213 

3279 

3344 

3409 

65 

6 

3474 

3539 

3605 

3670 

3735 

3800 

3865 

3930 

3996 

4061 

65 

7 

4126  4191 

4256 

4321 

4386 

4451 

4516 

4581 

4646 

4711 

65 

8 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

5296 

5361 

65 

9 

5426 

5491 

5556 

5621 

5686 

5751 

5815 

5880 

5945 

6010 

66 

670 

826075 

826140 

326204 

826269 

326334 

326399 

326-16-1 

826528 

826593 

826658 

66 

1 

6723 

6787 

6852 

6917 

6981 

7046 

7111 

7175 

7240 

7305 

65 

2 

7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

7951 

65 

3 

8015 

8080 

8144 

8209 

8273 

8338 

8402 

8467 

8531 

8595 

64 

4 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

6-1 

5 

9304 

9368 

9432 

9497 

9561 

9625 

9690 

9754 

9618 

9882 

64 

6 

9947 

830011 

330075 

830139 

330204 

830268 

830332 

330396 

830460 

830525 

64 

7 

830589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

64 

8 

1230 

1294 

1358 

1422 

1486 

1550 

1614 

1678 

1742 

1806 

64 

9 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

64 

680 

832509 

832573 

832637 

832700 

832764 

832328 

832892 

832956 

833020 

833083 

64 

1 

3147 

3211 

3275 

3338 

3402 

3466 

3530 

3593 

3657 

3721 

64 

2 

3784 

3848 

3912 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

64 

3 

4421 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

64 

4 

5056 

5120 

5183 

5247 

5310 

5373 

5437 

5500 

5564 

5627 

63 

5 

5691 

5754 

5817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

63 

6 

6324 

6387 

6451 

6514 

6577 

6641 

6704 

6767 

6830 

6894 

63 

7 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

63 

8 

7588 

7652 

7715 

7778 

7841 

7904 

7967 

8030 

8093 

8156 

63 

9 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

63 

690 

838849 

833912 

833975 

839038 

839101 

839164 

839227 

839289 

839352 

839415 

63 

l|  9478 

9541 

9604!  9667 

9729 

9792 

9855 

9918 

9981 

840043 

63 

21840106 

840169 

840232840294 

840357 

840420 

840482 

840545 

840608 

0671 

63 

3 

0733 

0796 

0859  0921 

0984 

1046 

1109 

1172 

1234 

1297 

63 

4 

135S 

1422 

1485'  1547 

161C 

1672 

1735 

1797 

1860 

1922 

63 

e 

193£ 

2047 

2110;  2172 

223S 

2297 

2360 

2422 

2484 

2547 

62 

6 

2602 

2672 

2734 

2796 

285S 

2921 

2983 

3046 

3108 

3170 

62 

7 

3235 

3295 

3357 

3420 

3482 

3544 

3606 

3669 

373 

3793 

62 

£ 

385£ 

3918 

398C 

4042 

4104 

4166 

4229 

4291 

4353 

4415 

62 

8 

4475 

4539 

4601 

4664 

4726 

478S 

4850 

4912 

4974 

5036 

62 

Ho.  0 

1 

a 

3 

4 

5 

6 

7 

8 

e 

Dtff. 

222 


TABLE    I.      LOGARITHMS     OF     NUMBERS. 


Mo. 

O 

1 

2 

3 

4 

5 

6 

7 

8  i  9 

Dlff. 

700 

345098 

345160 

845222 

845284 

845346 

845408 

345470 

845532 

845594:845656 

~62 

1 

5718 

5780 

5842 

5904 

5966 

6028 

6090 

6151 

6213 

6275 

62 

2 

6337 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

62 

3 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

62 

4 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

8066 

812& 

62 

5 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

8620 

8682 

8743 

62 

6 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

9358 

61 

7 

0419 

9481 

9542 

9604 

9665 

9726 

~9788 

9849 

9911 

9972 

61 

8 

850033 

350095 

850156 

850217 

850279 

85034(1 

850401 

850462 

850524 

850585 

61 

9 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

61 

710 

851258 

851320 

851381 

851442 

851503 

851564 

851625 

851686 

851747 

851809 

61 

.  1 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

61 

2 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

30*9 

61 

3 

3090 

3150 

3211 

3272 

3333 

3394 

3455 

3516 

3577 

3637 

61 

4 

3698 

3759 

3820 

3881 

3941 

4002 

^063 

4124 

4185 

4245 

61 

6 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

61 

6 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

61 

7 

5519 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

61 

8 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

60 

9 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

60 

720 

857332 

857393 

857453 

857513 

857574 

857634 

857694 

857755 

857815 

857875 

60 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

60 

2 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

60 

3 

9138 

9198 

9258 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

60 

4 

9739 

9799 

9859 

9918 

9978 

860038 

860098 

860158 

860218 

860278 

60 

6 

860338 

860398 

860458 

860518 

860578 

0637 

0697 

0757 

0817 

0877 

60 

6 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

60 

7 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

60 

8 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

60 

9 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3144 

3204 

3263 

60 

730 

863323 

863382 

863442 

863501 

863561 

863620 

863680 

863739 

863799 

863858 

69 

3917 

3977 

4036 

40% 

4155 

4214 

4274 

4333 

4392 

4452 

59 

J 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

69 

£ 

5104 

5163 

5222 

52S2 

5341 

5400 

5459 

5519 

5578 

5637 

59 

4 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

69 

6 

6287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

69 

6 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

69 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

69 

8 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8586 

69 

9 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

9173 

59 

740 

869232 

869290 

869349 

869408 

869466 

869525 

869584 

869642 

869701 

869760 

59 

9818 

98771  9935 

9994 

870053 

870111 

870170 

870228 

870287 

870345 

69 

I 

870404 

870462870521 

870579 

0638 

0696 

0755 

0813 

0872 

0930 

68 

3 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

58 

4 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2098 

58 

5 

2156 

22i6 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

68 

6 

2739 

2797 

2855 

2913 

2972 

3030 

3088 

3146 

3204 

3262 

68 

; 

3321 

3379  3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

68 

i 

3902 

39601  4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

68 

I 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4945 

5003 

68 

760 

875061 

875119 

875177 

875235 

875293 

875351 

875409 

875466 

875521 

875582 

ts 

5640 

5698  5756 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

58 

6218 

6276  6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

68 

] 

6795 

6853  6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

68 

, 

7371 

7429  7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

68 

, 

7947 

8004   8062 

8119 

8177 

8234 

8292 

8349 

.8407 

8464 

57 

i 

8522 

8579   8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

57 

9096 

9153   9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

57 

9669 

9726  9784 

9841 

9898 

9956 

880013 

880070 

880127 

880185 

67 

880242 

880299  880356 

880413 

880471 

880528 

0585 

0642 

0699 

0756 

67 

No 

O 

i  i  a 

3 

4 

6 

6 

7 

8 

9 

DIft 

TABLE    I.      LOGARITHMS    OF    NUMBERS. 


223 


No. 

0 

1 

« 

3 

4 

5 

6 

7 

8 

9 

Dlff. 

760 

880814 

880871 

880928 

880985 

881042 

881099 

881156 

881213 

881271 

881328 

~67 

1 

1385 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

57 

2 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

57 

3 

2525 

2581 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3037 

57 

4 

3093 

3150 

3207 

3264 

3321 

3377 

3434 

3491 

3548 

3605 

57 

6 

3661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4115 

4172 

57 

6 

4229 

4285 

4342 

4399 

4455 

4512 

4569 

4625 

4682 

4739 

57 

1  7 

4795 

4852 

4909 

4965 

5022 

5078 

5135 

5192 

5248 

5305 

57 

l  8 

5361 

5418 

5474 

5531 

5587 

5644 

5700 

5757 

5813 

5870 

57 

9 

5926 

5983 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

56 

770 

866491 

886547 

886604 

886660 

886716 

886773 

886829 

886885 

886942 

886998 

66 

1 

7054 

7111 

7167 

7223 

7280 

7336 

73921  7449 

7505 

7561 

56 

2 

7617 

7674 

7730 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

56 

3 

8179 

8236 

8292 

8343 

8404 

8460 

8516 

8573 

8629 

86S5 

56 

4 

8741 

8797 

8853 

8909 

8965 

9021 

9077   9134 

9190 

9246 

66 

5 

9302 

9358 

9414 

9470 

9526 

9582 

9638 

9694 

9750 

9806 

56 

6 

9862 

9918 

9974 

890030 

890086 

890141 

890197 

890253 

890309 

890365 

56 

7 

890421 

890477 

890533 

0589 

0645 

0700 

0756 

0812 

0868 

0924 

56 

8 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

56 

9 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

56 

780 

892095 

892150 

892206 

892262 

892317 

892373 

892429 

892484 

892540 

892595 

56 

1 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

56 

2 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

56 

3 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

55 

4 

4316 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

4814 

55 

5 

4870 

4925 

4980 

5036 

5091 

5146 

5201 

5257 

5312 

5367 

65 

6 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

55 

7 

5975 

6030 

6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

55 

8 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

55 

9 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

56 

790 

897627 

897682 

897737 

897792 

897847 

897902 

897957 

898012 

898067 

898122 

55 

1 

8176 

8231 

8286 

8341 

8396 

M51 

8506 

8561 

8615 

8670 

55 

2 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

55 

3 

9273 

9323 

9383 

9437 

9492 

9547 

9602 

9656 

9711 

9766 

65 

4 

9821 

9875 

9930 

9985 

900039 

900094 

900149 

900203 

900258 

900312 

55 

5 

900X7 

900422 

900476 

900531 

0586 

0640 

0695 

0749 

0804 

0859 

55 

6 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

55 

7 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

54 

8 

2003 

2057 

2112 

2166 

222! 

2275 

2329 

2384 

2438 

2492 

54 

9 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

54 

800 

903090 

903144 

903199 

903253 

903307 

903361 

903416 

903470 

903524 

903578 

54 

1 

3633 

3687 

•3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

54 

2 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

54 

3 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

5148 

5202 

54 

4 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

54 

5 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

54 

6 

6335 

63S9 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

54 

7 

6874 

6917 

6981 

7035 

7039 

7143 

7196 

7250 

7304 

7358 

54 

8 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

54 

9 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

54 

810 

908485 

908539 

908592 

908646 

908699 

908753 

908807 

908860 

908914 

908967 

54 

1 

9021 

9074 

9128 

9181 

9235 

9289 

9342 

9396 

9449 

9503 

54 

2 

9556 

9610 

9663 

9716 

9770 

9823 

9877 

9930 

9984 

910037 

53 

3 

910091 

910144 

910197 

910251 

910304 

910358 

910411 

910464 

910518 

0571 

53 

4 

0624 

0678 

0731 

0784 

0838 

0891 

0944 

0998 

1051 

1104 

53 

5 

1158 

1211 

1264 

1317 

1371 

1424 

1477 

1530 

1584 

1637 

53 

6 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2116 

2169 

53 

7 

2222 

2275 

2328 

2381 

2435 

2488 

2541 

2594 

2647 

2700 

63 

8 

2753 

2806 

2859 

2913 

2966 

3019 

3072 

3125 

3178 

3231 

63 

9 

3284 

3337 

3390 

3443 

3496 

3549 

3602 

3655 

3708 

3761 

53 

No. 

0 

1 

3 

3 

Dltf. 

224 


TABLE    I.      LOGARITHMS    OF     NUMIUIRS. 


£1 

913814 
4343 

913367 
43% 

913920 
4449 

913973 

45(12 

914026 
4555 

914079 
4608 

914132 
4660 

914I84J914237 
4713      4766 

914290     63 
4819     53 

2 

4872 

4925 

4977 

6030 

6083 

5136 

5i-'j 

5241 

5294 

5347     63 

3 

5400 

5453 

6505 

6558 

6611 

5664 

6716 

6769 

f-22      5875     53 

4 

6927 

5980 

6033 

6085 

6138 

6191 

6243 

62% 

6349      6401      63! 

6 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

6si2.     6-75      6927      53 

6 

6930 

7033 

7085 

713S 

7190 

7'243 

7295 

7348 

7400 

7453     63! 

7 

7506 

7658 

7611 

7663 

7716 

7768 

7820 

7873      7925 

797SI  '62 

8 

8030 

8083 

8135 

8188 

8240 

8293 

8345 

8397 

8450 

8502;    62 

9 

8565 

8607 

8659 

8712 

8764 

8816 

8ti69 

8921 

8973 

902C     68 

j 

830 

919078 

919130 

919183 

919235 

919287 

919340 

919392 

919444 

9194% 

91  9549'     68 

1 

9601 

%53 

9706 

9758 

9810 

0603 

9914 

9967 

020019 

92IKI71       53 

2 

180123 

920176 

920223 

y-rr-ii 

WKU 

92/13X4 

9204  :«• 

990488 

0541 

0593     62 

3 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114!     52 

4 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1630 

1582 

1634      63 

6 

1686 

1738 

1790 

1842 

1  -94 

1946 

1998 

2050 

2102 

2154'     6* 

6 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622      2674      6ft 

7 

2725 

2777 

2S29 

2—1 

2933 

•81 

3037 

m 

3140!     3litt      53; 

8 

3244 

32% 

3348 

3399 

3451 

3503 

3655 

3607 

3658J     3710     63 

9 

3762 

3S14 

3865 

3917 

3%9 

4021 

4072 

4124 

4176J     4228  i     69 

840 

924279 

924331 

924383 

924434 

9244JJ6 

924538 

924689 

924641 

924693924744*     5»' 

1 

47% 

4848 

IBM 

4951 

5(03 

6054 

6K« 

6157 

5209      6261      Bl 

2 

6312 

5364 

6415 

6467 

6518 

6670 

6621 

6673 

6725      6776      53) 

3 

6828 

6879 

6931 

6982 

6034 

Gn-f, 

6137 

6188 

6240!     6291 

51 

4 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

67f>4      6805 

51 

6 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7*r»      7319 

51 

6 

7370 

7422 

7473 

7524 

7676 

7627 

7678 

7730 

7781 

7832 

61 

7 

7883 

7936 

net 

8037 

BOB! 

8140 

8191 

8242 

8293 

8345 

61 

8 

83% 

8447 

MM 

8649 

8601 

8652 

8703 

8754 

8SOC- 

8857 

51 

0 

8908 

81 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9368 

61 

$60 
1 

929419 
9930 

J2yi7u 
9981 

92'.)  ",21 
JX»f.U 

'j-j;::.:^ 

930083 

'.i2'/;2.J 

930194 

929674 
000166 

92''.  ~' 
930236 

929776  WWWJ7 
9302K7  93<>33* 

929879 
93»0«9 

61 
61 

2 

930440 

93M91 

0642 

0592 

0643 

0694 

0746 

07% 

0647 

0898 

61 

3 

0949 

1000 

1061 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

61 

4 

1458 

1609 

1660 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

51 

6 

1966 

2017 

2063 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

51 

6 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

vn 

2930 

61 

7 

2931 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

33S6 

3437 

61 

8 

3487 

3538 

3689 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

61 

0 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

61 

860 

934498 

934549 

934599 

934660 

934700 

934751 

934801 

934852 

934902 

934953 

60 

5003 

6054 

6104 

6164 

6205 

6255      6306 

6356 

5406 

5457 

60 

2 

6507 

6558 

6603 

6658 

6709 

6759      6809 

6860 

6910      5%0 

60 

3 

6011 

6061 

6111 

6162 

6212 

6262     6313 

6363 

6413      6463 

60 

4 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916!     6966 

60 

6 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

74  IS      7468 

60 

6 

7618 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919      7969 

60 

7 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

;     8420 

6470 

60 

8 

8520 

8570 

6620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

60 

9 

9020;     9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

60 

870 

939519  939569  939619 
9400181940068940118 

939669 
940163 

939719 

940218 

939769939819 
940-267940317 

939869  939918 
940367  940417 

939968 
940467 

60 
50 

2 

0516     0566     0616 

0666 

0716 

07C5 

0815 

0,^65 

0915 

0964 

60 

3 

1014      1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412      1462 

60 

4 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909      l'J5S 

60 

5 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405      2455 

60 

6 

1504 

2554      2603     2653 

2702 

2752 

2801 

2851 

2901 

2950 

50; 

7 

3000     3.ns)     3.NM!    3148 

3193 

.   3217 

3297 

3346 

33% 

3445 

49! 

8 

3495      3514  i     35^8      3643 

3692 

3742 

3791 

3841 

3890 

3939 

49 

9 

3989;    4038     4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433;    49 

Wo 

0 

1 

a 

3 

| 

5 

6 

7 

i     8 

9 

Dlfl  i 

TAIU.I.      I.       l.OCAKITHMS     OF     NTMHEKS. 


No.'  0 

1 

8 

9 

Diff. 

880 

<141483 

W4532 

944531 

944631 

M46SO 

M4729 

944779 

944828 

944877 

944927 

49 

1 

4976 

5026 

5074 

5124 

5173 

5222J  5272 

5321 

6370 

5419 

49 

a 

5469 

55!  8 

6567 

5616 

6665 

5715 

6764 

6813 

6362 

5912 

49 

3 

5981 

6010 

6059 

6103 

6157 

6207 

6256 

6305 

6354 

6403 

49 

4 

6452 

6/501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

49 

6 

6943 

6992 

7041 

7090 

7140 

7189 

7238 

7287 

7336 

7385 

49 

0 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7876 

49 

7 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

49 

8 

8413 

8462 

8511 

8560 

8609 

8657 

8706 

8765 

8804 

8853 

49 

9 

8902 

8951 

8999 

9048 

9097 

9146 

9196 

9244 

9292 

9341 

49 

390! 

949390 

949439 

949488 

949536 

949585 

949634 

949683 

949731 

949780 

949829 

49 

I   9878 

9926 

9975 

950024 

950073 

50121 

950170 

950219 

950267 

950316 

49 

2!>o0365 

950414 

950462 

0511 

0560 

0608 

0657 

0706 

0754 

0803 

49 

3  0851 

0900 

0949 

0997 

1046 

1095 

111  : 

1192 

1240 

1289 

49 

41  1338 

13^6 

1435 

1483 

1532 

1580 

1629 

1677 

1726 

1775 

49 

6 

1823 

1S72 

1920 

1969 

2017 

2066 

*in 

2163 

2211 

2260 

48 

6 

2308 

2356 

24M5 

2453 

2502 

2550 

2599 

2647 

2696 

2744 

48 

7 

2792 

2841 

2889 

2938 

2936 

3034 

3083 

3131 

3180 

3228 

48 

8 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

48 

9 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

48 

900 

954243 

954291 

954339 

954387 

954435 

954484 

954532 

954580 

954628 

954677 

48 

I 

4725 

4773 

4S21 

4S69 

4918 

4966 

6014 

6062 

6110 

6158 

48 

2 

6207 

5255 

5303 

6351 

5399 

5447 

6495 

6543 

6592 

6640 

48 

3 

5688 

5736 

5784 

5832 

5380 

6928 

6976 

6024 

6072 

6120 

48 

4 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6653 

6601 

48 

6 

6C49 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

48 

6  7128 

7176 

7224 

7272 

7320 

7363 

7416 

7464 

7512 

7559 

48 

7|  .7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

48 

8 

8036 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8616 

48 

9 

8664 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

48 

910 

959041 

959089 

959137 

959185 

959232 

95923C 

969328 

959375 

959423 

959471 

48 

1 

US  1  8 

9566 

9614 

9661 

9709 

9757  9804 

9852 

9900 

9947 

48 

2 

9995 

960fU2 

960090 

9601  33 

960185 

960233  9602SO 

960328 

960376 

960423 

48 

3 

%0471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0861 

0899 

48 

4  0946 

0994 

1041 

1039 

1136 

1184 

1231 

1279 

1326 

1374 

48 

5  1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1849 

47 

6  1895 

1943 

1990 

2038 

2035 

2132 

2180 

2227 

2275 

2322 

47 

7  2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

47 

8l  2843 

2890 

2937 

2935 

3032 

3079 

3126 

3174 

3221 

3268 

47 

9  3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

47 

920963783 

963835 

963882 

963929 

963977 

964024 

964071 

9641  IS 

964165 

964212 

47 

1 

426C 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

47 

2 

473 

4778 

4825 

4372 

4919 

4966 

5013 

5061 

6108 

5155 

47 

3 

5202 

5249 

5296 

5343 

5390 

6437 

5484 

6531 

6678 

5625 

47 

4 

5672 

5719 

5766 

5313 

6S60 

5907 

5954 

6001 

6048 

6095 

47 

5 

6142 

6189 

6236 

6233 

6329 

6376 

6423 

6470 

6517 

6564 

47 

6 

661 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7033 

47 

7 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7454 

7501 

47 

8 

7548 

7595 

7642 

7638 

7735 

7782 

7829 

7875 

7922 

7969 

47 

9 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

47 

930 

968483 

968530 

968576 

968623 

968670 

963716 

968763 

968810 

968856 

968903 

47 

11  8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

47 

1 

9416 

9463 

9508 

9556 

960? 

9649 

9695 

9742 

9789 

9835 

47 

3 

9882 

9928 

9975 

970021 

970063 

9701  14 

970161 

970207 

970254 

970300 

47 

4 

97034 

970393 

970440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

46 

5 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

46 

6 

127 

1322 

1369 

1415 

146 

1508 

1554 

1601 

1647 

1693 

46 

7 

1740 

1786 

1832 

1879 

192o 

1971 

2018 

2064 

2110 

2157 

46 

8 

220 

2249.  2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

46 

0 

2666 

2712]  2768 

2304 

285 

2897 

2943 

2989 

3035 

3082 

46 

No.   0 

9 

Diff. 

TABLE    I.      LOGARITHMS    OF    NUMBERS. 


No. 

940  973128  973174  973220  973266  973313  973359  973405  973451  973497  973548 


950977724 


8181 
BOS? 


1 

1 
8 
4 

;,  9NNN..", 

t;  oi> 
7  0912 
1366 
1619 


4051 
4512 
4972 
5432 

••SIM 
6350 

•Wis 


2723 
3175 
8888 

4077 

4:,27 

J9i 7 
6488 

:,s;:, 
6324 

970986772 


7219 

roM 

8113 
BBBO 
BOOCS 


9696 

90339 

o;s:5 


2111 
2.V>1 
8906 
3436 
8877 
4317 
4757 
5196 


6074 
6512 


8259 


8  9131 

9  9565 

Noj     0 


4097 


5478 
5937 


7312 


977769  97781.-)  977861  977906  977962  977998  978043  978089  978135 


9BM 


980049980094 


0503 
09:,; 
1411 
1864 


2709 
3220 
3671 
4122 
4572 
r.n-,'2 
6471 


72»;» 
7711 
8157 
8604 
904U 
'.•I'.' I 
9989 


S  <>90339  990883  990428 

ON] 


980  «KM22T,.t!»1270  991315 


1713 
8136 


3480 
3921 
4:161 
4801 
5240 


611 
6555 


7867 


8739 
9174 


4143 
4604 

5m;  4 

5524 
6868 

6142 
6900 
7358 


8272 


9184 
9639 


0649 
1008 

i  »:,»; 

1909 


2814 
3365 
8716 
4167 
4617 
6097 
6616 
6868 
6418 


7809 

7758 
8809 

6646 

'.*>-.•  I 
9689 


1758 

221  HI 

2642 


4405 

4*45 


6161 
6599 
7037 
7474 
7910 
8347 
8782 
9218 
9652 


3728 
4169 

4650 
51101 

;V)7(i 

8089 
8466 
8846 
7408 


3774 
42*-.  4281 
4696!  4742 
6156  5202 
5616 


8076 

•1533 

t;w2 

7449 


8317 

8774  8819 


980140  980185  980881 


1114* 

1501 
1954 


8810 
8789 
4212 
4662 
6112 
6561 
6010 
6458 


0640 

lu'..:5 
1547 
2000 


290 » 

8868 

3NC 
42.".: 
4707 
6157 

.v  ;,*•, 

6066 

6608 


168905968951 
7:Vi3  7896 
7800.  7845 
8247  8291 

8693  8737 

9138  9183 

9672 

990028990072990117 
"i;2  0516  0561 
0916!  0960  1004 


6121 

e:,; '.i 
7037 
7495 


S4"9 

*N;:, 
9881 
97W 


1139 
1898 

201.-. 


.'iloi 

6868 
4808 

4;.vj 

.YJ02 

6851 

C,].N) 


?  I  J.", 


87X2 


1802 
2211 


8187 
8686 


4449 


990  995635  995679  995723  995767  995811  996654  995898  995942  99K966  996030 


6205 
6643 
7080 
7517 
7954 
S:«MI 


1846  1890 

2288  2338 

2730  2774 

8172  8216 

3613  365' 

4053  4097 

4493  4537 

4933  4977 

5372,  5416 


7124 
7561 
7998 
S43I 


6731 
7168 
7605 
8041 
8477 


8869  8913 
9305  9348 
9739  9783 


4327 

47!* 


6707 
6187 


7088 

7541 


8454 
8911 
9868 

8881 


y»uy 
0730 
1184 
1637 
8090 


(5  980322  980367  980412 


8448 

8897 
4847 

47  '.-7 

6847 

6696 
6144 


7984 


'.••.'72 

9717 

990161 

0605 

KM!) 


3913 
4374 

4*34 

5753 
6212 
6671 
7129 
7586 


6500 
6958 

9412 


0776 
1899 

16S3 
2135 


8481 

8942 
4998 

JM'J 

6898 

6741 
6189 


7588 

7'..;;. 
6496 
6871 
9816 
9761 


4420 
4880 
5340 
5799 
6856 
8717 
7175 
7632 


4005 
44  6 


6763 
7220 
7678 


K>4G 
9002 
U457 


0681 

127:. 
1728 
2181 


8586 

8987 
4437 

4-T 

6837 
6786 
6284 


7577 

N.-JJ 

6470 
6916 

«»3C,I 

9606 


0650J  0694 
1098!  1137 


991359  991403  991448  991492  991530  991580  991625 


8591 
9047 


9968 


0689 
1880 

1773 


8681 
4088 

41S2 

4932 

1888 

Win 

027  9 
6727 


KM! 


9406 
9660 


0738 
1182 


19a"j  1979 

2377  2421 

2819  2863 

3260  3304 

8701  374')' 

4141  4185 

4581  4625 

5021  5065: 

5460,  5504, 


2023 
2465 
2907 
3348 
8789 
4229 


2067 
2509 
29.r>l 


4273 


4Wi  4713 
5108;  6152 
5547!  6591 


6337,  6380 

6774  6818, 

7212'  7255 

7648  7692 

8085!  8129 

8581  8564! 

8956;  9000! 

9392  9435 

9826  9870 

~6|  7     1 


6424  6468 

6862  6906 

7299|  7343 

7736  7779 

sS  £"•-•' 

9043|  9087 

9479!  9522 

9913  9957 


Diff. 


44 
41 
44 

41 
41 
41 
14 
41 
41 
J3 

Diff. 


TABLE  II. 

LOGARITHMIC  SINES,  COSINES,  TANGENTS  AND 
COTANGENTS. 


227 


TABLE    II.       LOGARITHMIC    SI 


NOTE. 

THE  table  here  given  extends  to  minutes  only.  The  nsual 
method  of  extending  such  a  table  to  seconds,  by  proportional 
parts  of  the  difference  between  two  consecutive  logarithms,  is  ac- 
curate enough  for  most  purposes,  especially  if  the  angle  is  not 
very  sinalL  When  the  angle  is  very  small,  and  great  accuracy  is 
required,  the  following  method  may  be  used  for  sines,  tangents, 
and  cotangents. 

I.  Suppose  it  were  required  to  find  the  logarithmic  sine  of  5'  24  . 
By  the  ordinary  method,  we  should  have 

log.  sin.  5          =  7.162690 
diff.  for  24 '      =      31673 


log.  sin.  5'  24"  =  7. 

The  more  accurate  method  is  founded  on  the  proposition  in  Trigo- 
nometry, that  the  sines  or  tangents  of  very  small  angles  are  pro- 
portional to  the  angles  themselves.  In  the  present  case,  there- 
fore, we  have  sin.  5' :  sin.  5  24'  =5:5  24'  =  300  : 324' .  Hence 

sin.  5' 24    =  324^°'°,  or  log.  sin.  5'  24    =  log.  sin.  5  +  log.  324  - 
tJUU 

log.  300.  The  difference  for  24"  will  therefore,  be  the  difference 
between  the  logarithm  of  324  and  the  logarithm  of  300.  The 
operation  will  stand  thus : — 

log.  324  =  2.510545 

log.  300  =  2.477121 

diff.  for  24 '      =      33424 
log.  sin.  5'         =  7.162696 

log.  sin.  5  24    =  7.196120 

Comparing  this  value  with  that  given  in  tables  that  extend  to 
seconds,  we  find  it  exact  even  to  the  last  figure. 

IL  Given  log.  sin.  A  =  7.004438  to  find  A.    The  sine  next  less 
than  this  in  the  table  is  sin.  3'  =  6.940847.    Now  we  have  sin.  3  : 


AXD  OOTJLXGE3GTR.  '_*_ 

log.  sin. -1  —  log.  sin.  3.  Henoe  it  appeals,  that,  to  find  the  loga- 
rithm of  A  in  minutes,  we  must  add  to  the  logarithm  of  3  the 
difference  between  log.  sin.  A  and  log.  sin.  3". 

log.  sin.  A  =  7.004438 
log.  an.  V  =  61940847 

MU 

kg.  3          =  0.477121 

A  =  3.473       0510712 

or  A  =  3*  3&38  .  By  the  common  method  we  should  hare  found 
A  =  3  30.54  . 

The  same  method  applies  to  tangents  and  cotangents,  except 
that  in  the  case  of  cotangents  the  differences  are  to  be  subtracted. 


%  *  The  radios  of  this  table  is  unity,  and  the 
8,  7,  and  6  stand  respectively  lor  — t,  —2,  —3,  and 


230 


TABLE    II.       LOGARITHMIC    SINES, 


M. 

Sine. 

D.I  . 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

11. 

1 
2 
3 
4 

5 

Inf.  neg. 
6.463726 
.764756 
.940847 
7.065786 
.162696 

5017.17 

2934.85 
2082.31 
1615.17 

0.000000 
.000000 
.000000 
.000000 
.000000 
.000000 

.00 
.00 
.00 
.00 
.00 

r*n 

Inf.  neg. 

6.463726 
.764756 
.940847 
7.065786 
.162696 

5017.17 
2934.85 
2((82.31 
161517 

Infinite. 

3.536274 
.235244 
.059153 
2.934214 
.837304 

60 
59 
58 
67 
56 
65 

6 

7 

.241877 
.308824 

1I15>8 

Qfifi  W 

9.999999 
999999 

.uu 
.00 

HA 

.241878 
.308825 

1319.69 
1115.78 

n/»£  &A 

.758122 
.691175 

54 
63 

8 

.366S16 

yoo.oo 

999999 

.uu 

.366817 

yt>o.o4 

QCO  CC 

.633183 

62 

9 

.417968 

762.62 

.999999 

.01 

.417970 

OO.4.55 
762.63 

.582030 

61 

10 

7.463726 

fiQQ  QQ 

9.999998 

111 

7.463727 

AQQ  QQ 

2.536273 

50 

11 
12 
13 
14 
15 
16 
17 
18 
19 

.505118 
.542906 
.577668 
.609853 
639816 
.667845 
.694173 
.718997 
.742478 

Doy.oo 

629.81 
579.37 
536.  4  j 
499.38 
467.14 
438.81 
413.72 
391.35 
371.27 

.999998 
999997 
.999997 
.999996 
.999996 
999995 
.999995 
999994 
999993 

.Ul 

.01 
.01 
.01 
.01 
.01 
.01 
.01 
.01 
.01 

.505120 
.542909 
.577672 
.609857 
639820 
.667849 
.694179 
.719003 
.742484 

DCHP.OO 

629.81 
579.37 
536.42 
499.39 
467.15 
438.82 
413.73 
391.36 
371.28 

.494880 
.457091 
.422328 
.390143 
.360180 
.332161 
.305821 
.280997 
.257516 

49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 

7.764754 
.785943 
806146 
.825451 

353.15 
336.72 
321.75 

QflQ  ft* 

9.999993 
999992 
999991 
999990 

.01 
.01 
.01 

7.764761 
.785951 
.806155 
.825460 

353.16 
336.73 
321.76 

QAQ  H7 

2.235239 
.214049 
193845 
.174540 

40 
39 
38 
37 

24 
25 

.843934 
.861662 

oUo.Uo 

295.47 

999989 
999989 

02 
OQ 

.843944 
861674 

oUo.U/ 

295.49 

ooo  on 

.156056 
138326 

36 
35 

26 

.878695 

273!  17 

999988 

.u* 
.02 

.878708 

-coo.yu 
273.18 

121292 

34 
oo 

28 

29 

'.910879 
.926119 

263.23 
253.99 
245.38 

999987 
999986 
999985 

02 
.02 
.02 

.910894 
926134 

263.25 
254.01 
245.40 

1089106 

00 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

7.940842 
.955082 
.968870 
.982233 
.995198 
S.007787 
.020021 
.031919 
.043501 
.054781 

237.33 
229.80 
222.73 
216.08 
209.81 
203.90 
198.31 
193.02 
188.01 
183.25 

9.999983 
999982 
999981 
999980 
999979 
.999977 
.999976 
.999975 
.999973 
999972 

.02 

.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 

7.940858 
.955100 
968889 
982253 
.995219 
8.007809 
.020044 
.031945 
.043527 
.054809 

237.35 
229.82 
222.75 
216.10 

209.83 
203.92 
19833 
193.05 
188.03 
183.27 

2.059142 
.044900 
.031111 
.017747 
.004781 
1.992191 
.979956 
.968055 
.956473 
.945191 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

8.065776 
.076500 
.086965 
.097183 
.107167 
.116926 
.126471 
.135810 
.144953 
.153907 

178.72 
174.42 
170.31 
166.39 
162.65 
159.08 
155.66 
152.38 
149.24 
146.22 

9.999971 
.999969 
999968 
.999966 
.999964 
.999963 
.999961 
.999959 
.999958 
.999956 

.02 
.03 
.03 
.03 
.03 
.03 
.03 
.03 
.03 
.03 

8.065806 
.07653! 
.086997 
.097217 
.107203 
116963 
.126510 
.135851 
144996 
.153952 

178.75 
17444 
170.34 
16642 
162.68 
159.11 
155.69 
15241 
149.27 
146.25 

1.934194 
.923469 
.913(103 
.902783 
.892797 
.883037 
.873490 
.864149 
.855004 
.846048 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

8.162681 
.171280 
.179713 
.187985 
.196102 
.204070 
.21  1895 
.219581 
.227134 
.234557 
.241855 

143.33 
140.54 
137.86 
135.29 
132.80 
130.41 
128.10 
125.87 
123.72 
121.64 

9.999954 
.999952 
999950 
.999948 
999946 
999944 
999942 
999940 
999938 
999936 
.999934 

.03 
.03 
.03 
.03 
.03 
.03 
.03 
.04 
.04 
.04 

8.162727 
.171328 
.179763 
.188036 
.196156 
.204126 
.211953 
.219641 
227195 
.234621 
241921 

14336 
140.57 
13790 
135.32 
132.84 
13044 
129.14 
125.91 
123.76 
121.68 

1.837273 
.828672 
.820237 
.811964 
.803844 
.795874 
.788047 
.780359 
.772805 
.765379 
.768079 

10 
9 
8 
7 
6 
6 
4 
3 
2 

0 

M. 

Cosine. 

D.  1". 

Biaa. 

».  1". 

Oofiaug. 

D.  l'». 

Tang. 

M. 

89° 


,    TANGENTS,    AND   COTANGENTS. 


M. 

Sine 

D.  !«•. 

Cceine. 

D  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

8.241855 
.243033 

119.63 

9.999934 
.999932 

.04 

8.241921 
.249102 

119.67 

1.758079 
.750898 

lib" 

59 

.256094 
.263042 
.269881 
.276614 
.283243 

117.69 
115.80 
113.98 
112.21 
110.60 

.999929 
.999927 
.999925 
.999922 
.999920 

.04 
.04 
.04 
.04 
.04 

.256165 
.263115 
.269956 
.276691 
.283323 

1  17.72 
115.84 
114.02 
112.25 
110.54 

.743835 
.736885 
.730044 
.723309 
.716677 

58 
57 
56 
55 
54 

.289773 
.296207 
.302546 

108.83 
107.22 
105.66 
104.13 

.999918 
.999915 
.999913 

.04 
.04 
.04 
.04 

.289856 
.29S292 
.302634 

108.87 
107.26 
105.70 
104.18 

.710144 
.703708 
.697366 

53 
52 
51 

10 
U 
12 
13 
14 
15 
16 
17 
13 
19 

8.308794 
.314954 
.321027 
.327016 
.332924 
.338753 
.344504 
.350181 
.355783 
.361315 

102.66 
101.22 
99.82 
98.47 
97.14 
95.86 
94.60 
93.38 
92.19 
91.03 

9,99*910 
.999907 
.999905 
.999902 
.999899 
.999897 
.999894 
.999891 
.999888 
.999885 

.04 
.04 
.04 
.05 
.05 
.05 
.05 
.05 
.05 
.05 

8.308884 
.315046 
.321122 
.327114 
.333025 
.338856 
.344610 
.350289 
.3C5895 
.361430 

102.70 
101.26 
99.87 
98.51 
97.19 
95.90 
94.65 
93.43 
92.24 
91.08 

1.691116 
.684954 
.678878 
672886 
666975 
.661144 
.655390 
.649711 
.644105 
.638570 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 

8.366777 

9.999882 

AC 

8.366395 

1.633105 

40 

21 

.372171 

QQ  Qf\ 

.999879 

tvO 

.372292 

89.95 

.627708 

39 

23 
24 
26 
26 
27 

29 

.377499 
.382762 
.387962 
.393101 
.398179 
.40311)9 
.408161 
.413063 

oo.oU 
87.72 
86.67 
85.64 
84.64 
83.66 
82.71 
81.77 
80.86 

.999876 
.999873 
.999870 
.999867 
.999864 
.999861 
.999858 
.999854 

'.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 

.377622 
.382889 
.388092 
.393234 
.398315 
.403338 
.408304 
.413213 

88.85 
87.77 
86.72 
85.70 
84.69 
83.71 
82.78 
81.82 
80.91 

.622378 
.617111 
.611908 
.606766 
.601685 
.596662 
.591696 
.686787 

38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
36 
38 
37 
38 
39 

8.417919 
.422717 
.427462 
.432156 
.436800 
.441394 
.445941 
.450440 
.454893 
.459301 

79.96 
79.09 
78.23 
77.40 
76.58 
75.77 
74.99 
74.22 
73.47 
72.73 

9.999851 
.999848 
.999844 
.999841 
.999833 

1999831 
.999827 
.999824 
.999820 

.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 

8.418068 
.422869 
.427618 
.432315 
.436962 
.441560 
.446110 
.450613 
.455070 
.459481 

80.02 
79.14 
78.29 
77.45 
76.63 
75.83 
75.05 
74.28 
73.53 
72.79 

1.581932 
.677131 
.572382 
.567685 
.563038 
558440 
.653890 
.549387 
.544930 
.540519 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 

8.463665 
.467985 
.472263 
.476498 

72.00 
71.29 
70.60 

9.999816 
.999813 
.999809 
.999805 

.06 
.06 
.06 
rut 

8.463849 
.468172 
.472454 
.476693 

72.06 
71.35 
70.66 

1.536151 
.531828 
.527546 
.523307 

20 
19 
18 
17 

44 

.480693 

69.91 

.999801 

.Uo 
ne 

.480892 

69.98 

.519108 

16 

45 

46 
47 

48 
49 

.484848 
488963 
.493040 
.497078 
.501080 

68^59 
67.94 
67.31 
66.69 
66.08 

.999797 
.999794 
.999790 
.999786 
.999782 

.UB 
.06 
.07 
.07 
.07 
.07 

.4,85050 
.489.70 
.493250 
.497293 
.501298 

68!65 
68.01 
67.38 
66.76 
66.15 

.514950 
.510830 
.506750 
.502707 
.498702 

15 
14 
13 
12 
11 

50 

ft  505045 

9  999778 

AT 

8.505267 

CC  EC  ' 

1.494733 

10 

51 

.508974 

65.48 

.999774 

.U/ 

AT 

.509200 

QQUOO 

.490800 

9 

52 
53 

.512867 
.516726 

64.32 

CO  7C 

.999769 
.999765 

.U7 
.07 

AT 

.513098 
.516961 

64.39 

CO  QO 

.486902 
.483039 

8 
7 

54 

55 

.520551 
.524343 

oo.  7o 

63.19 

.999761 
.999757 

.tl/ 

.07 

AT 

.520790 
.524586 

DO.O4 

63.26 

.479210 
.475414 

6 
5 

58 

.528102 

CO  |  | 

.999753 

.07 

AT 

.528349 

62.72 

CO  I  Q 

.471651 

4 

57 

58 
59 
80 

.531328 
.535523. 
.539186 
.542819 

(XG.  I  ! 

61.58 
61.06 
60.55 

.999748 
.999744 
.999740 
.999735 

.U/ 

.07 
.07 
.07 

.532080 
.535779 
.539447 
.543084 

O4.  lo 

61.65 
61.13 
60.62 

.467920 
.464221 
.460553 
.456916 

3 
2 

0 

M. 

Corfne. 

D.I'. 

Sine. 

D.  1". 

Cotang. 

P.  1". 

Tang. 

M. 

910 


88* 


232                              TABLE    II.       LOGARITHMIC    SINES. 
8°                                                                                                            Iff3 

M. 

Sine. 

D.  1". 

Cosine. 

D  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

8.542819 
.546422 

60.04 

9.999735 
.999731 

.07 

8.543084 
.546691 

60.12 

1.456916 
.453309 

60 
59 

2 
3 

.549995 
.553539 

59.06 

.999726 
.999722 

.08 

.550268 
.553817 

69.14 

.449732 
.446183 

68 
67 

4 

.657054 

.999717 

.557336 

.442664 

66 

6 
6 

8 
9 

.560540 
.563999 
.567431 
570836 
.574214 

58.11 
67.65 
57.19 
56.74 
56.30 
55.87 

.999713 
.999708 
.999704 
999699 
.999694 

.08 
.08 
.08 
.08 
.08 

.560828 
.564291 
567727 
571137 
.574520 

57.73 
57.27 
66.82 
56.38 
65.95 

.439172 
.435709 
.432273 
.428863 
.425480 

55 
64 
53 
62 
61 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

8.577566 
.580892 
.584193 
.587469 
.590721 
.593948 
.597152 
.600332 
.603489 
.606623 

55.44 
55.02 
54.60 
54.19 
53.79 
53.39 
53.00 
52.61 
52.23 
51.86 

9.999689 
.999685 
.999680 
999675 
999670 
999665 
.999660 
.999655 
.999650 
.999645 

.08 

.08 
.08 
.08 
.08 
.08 
.08 
.08 
.08 
.09 

8.677877 
.681208 
.684514 
587795 
.591051 
.594283 
.597492 
.600677 
.603839 
.506978 

55.52 
55.10 
64.68 
64.27 
63.87 
53.47 
53.08 
52.70 
52.32 
51.94 

1.422123 
.418792 
.415486 
.412205 
.408949 
.405717 
.402508 
,399323 
.396161 
.393022 

60 
49 
48 
47 
46 
46 
44 
43 

41 

20 

8.609734 

9.999640 

8.610094 

1.389906 

40 

21 

99 

.612823 

51.12 

.999635 

.09 

.613189 

51.21 

.386811 

39 

OQ 

23 
24 
25 
26 
27 
28 
29 

.618937 
.621962 
.624965 
.627948 
.630911 
.633854 
.636776 

60.77 
60.41 
50.06 
49.72 
49.38 
49.04 
48.71 
48.39 

.999624 
.999619 
.999614 
.999608 
.999003 
.999597 
.999592 

.09 
.09 
.09 
.09 
.09 
.09 
.09 
.09 

.619313 
.622343 
.625352 
.628340 
.631308 
.634256 
.637184 

50.85 
50.50 
50.15 
49.81 
49.47 
49.13 
48.80 
48.48 

380687 
377657 
.374648 
.371660 
.368692 
.365744 
.362816 

37 
35 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

8.639680 
.642563 
.645428 
.648274 
.651102 
.653911 
.656702 
.659475 
.662230 
.664968 

48.06 
47.75 
47.43 
47.12 
46.82 
46.52 
46.22 
45.93 
45.63 
45.35 

9.999586 
.999581 
.999575 
.999570 
.999564 
.999558 
.999553 
.999547 
.999541 
.999535 

.09 
.09 
.09 
.09 
.09 
.10 
.10 
.10 
.10 
.10 

8.640093 
.642982 
.645853 
.648704 
.651537 
.654352 
.657149 
.659928 
.662689 
.665433 

48.16 
47.84 
47.53 
47.22 
46.91 
46.61 
46.31 
46.02 
45.73 
45.45 

1.359907 
.357018 
.354147 
.351296 
.348463 
.345648 
.342851 
.340072 
.337311 
.334567 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

8.667689 
.670393 
.673080 
.675751 
.678405 
.681043 
.683665 
.686272 
.688863 
.691438 

45.07 
44.79 
44.51 
44.24 
43.97 
43.70 
43.44 
43.18 
42.92 
42.67 

9.999529 
.999524 
.999518 
.999512 
.999506 
.999500 
.999493 
.999487 
.999481 
.999475 

.10 
.10 
.10 
.10 
.10 
.10 
.10 
.10 
.10 
.10 

8.668160 
.670870 
.673563 
.676239 
.678900 
.681544 
.684172 
.686784 
.689381 
.691963 

45.16 
44.88 
44.61 
44.34 
44.07 
43.80 
43.64 
43.28 
43.03 
42.77 

1.331840 
.329130 
.326437 
•323761 
.321  100 
.318456 
.315828 
.313216 
.310619 
.308037 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
61 
52 
53 
54 
65 
66 
57 
58 
59 

8.693998 
.696543 
.699073 
.701589 
.704090 
.706577 
.709049 
,711507 
.713952 
.716383 

42.42 
42.17 
41.93 
41.68 
41.44 
41.21 
40.97 
40.74 
40.51 

9.999469 
.999463 
.999456 
.999450 
.999443 
.999437 
.999431 
.999424 
.999418 
.999411 

.10 

.11 
.11 
.11 
.11 
.11 
.11 
.11 
.11 

8.694529 
.697081 
.699617 
.702139 
.704646 
.707140 
.709618 
.712083 
.7H534 
.710972 

42.52 
42.28 
42.03 
41.79 
41.65 
41.32 
41.08 
40.85 
40.62 
4O  40 

1.305471 
.302919 
.300383 
.297861 
.295354 
.292860 
.290382 
.287917 
.285466 
.283028 

10 
9 

8 

7 

5 
4 
3 
a 

60 

.718800 

.999414 

.719396 

.280604 

0 

M. 

Cosine. 

D.  1". 

Sine         D.I". 

Cotang. 

Tang. 

M. 

COSINES,    TANGENTS,    AND   COTANGENTS. 


233 


30 


M. 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

1 
2 

8.718800 
.721204 
.723595 

40.06 
39.84 

OQ  CO 

9.999404 
.999393 
.999391 

.11 
.11 

8.719396 
.721806 
.724204 

40.17 
39.95 

1.280604 
.278194 
.275796 

60 
59 

58 

3 
4 

.725972 
.728337 

•VwCHI 

39.41 

on  i  Q 

.999384 
.999378 

.11 

.726588 
.728959 

3^52 

OQ  qi 

.273412 
.271041 

57 
56 

5 

.730688 

oy.  i  y 

OQ  QQ 

.999371 

.731317 

oy.oi 

.268683 

55 

6 

.733027 

oo.yo 

OQ  77 

.999364 

.733663 

OQ  Q 

.266337 

54 

7 

.735354 

oo.  /  / 

OQ  C7 

.999357 

.735996 

00*  CO 

.264004 

53 

8 

.737667 

OO.O/ 

.999350 

10 

.738317 

OO.OO 

.261683 

52 

9 

.739969 

sa  16 

999343 

.  1<£ 

.12 

.740626 

38.'27 

.259374 

51 

10 
11 
12 

3.742259 
.744536 
.746802 

37.96 
37.76 

07  cc 

9.999336 
.999329 
.999322 

.12 
.12 

8.742922 
.745207 
.747479 

38.07 

37.88 

Q7  CQ 

1.257078 
.254793 
.252521 

50 
49 
43 

13 

14 
15 

.749055 
.751297 
.753528 

o/.oo 
37.37 
37.17 

.999315 
.999308 
.999301 

!l2 
.12 

.749740 
.751989 

.754227 

O/.OO 

37.49 
37.29 

07  m 

.250260 
.24801  1 
.245773 

47 
46 
45 

16 
17 

.755747 
.757955 

36.'80 

.999294 
.999237 

iia 

1  O 

.756453 
.758668 

oY.lU 
36.92 

OC  ~"> 

.243547 
.241332 

44 
43 

18 
19 

.760151 
.762337 

36.61 
36.42 
36.24 

.999279 
999272 

.1* 
.12 
.12 

.760872 
.763065 

ob.76 
36.55 
36.36 

.239128 
.236935 

42 
41 

20 

21 

8.764511 
.766675 

36.06 

9599265 
.999257 

.12 

8.765246 
.767417 

36.13 
ofi  nn 

1.234754 
.232583 

40 
39 

22 
23 
24 
25 
26 
27 
23 
29 

.768828 
.770970 
.773101 
.775223 
.777333 
.779434 
.781524 
783605 

35!  70 
35.53 
35.35 
35.18 
35.01 
34.84 
34.67 
34.51 

.999250 
.999242 
.999235 
.999227 
999220 
999212 
999295 
999197 

!l2 
.12 
.13 
.13 
.13 
.13 
.13 
.13 

.769578 
.771727 
.773866 
.775995 
.778114 
.780222 
.782320 
.784408 

oo.UU 
35.83 
35.65 
35.48 
35.31 
35.14 
34.97 
34.80 
34.64 

.230422 
.228273 
.226134 
.224005 
.221886 
.219778 
.217680 
.215592 

38 
37 
36 
36 
34 
33 
32 
31 

30 
31 
32 

8.785675 
.787736 
.789787 

34.34 
34.18 

9.999189 
999181 
999174 

.13 
.13 

8.786486 
.788554 
.790613 

34.47 
34.31 

1.213514 
.211446 
.209387 

30 
29 

23 

33 
34 
35 
36 
37 
33 

.791828 
.793859 
.795881 
.797894 
.799897 
.801892 

sase 

33.70 
33.54 
33.39 
33.23 
oo  no 

.999166 
.999153 
.999150 
.999142 
.999134 
.999126 

.13 
.13 
.13 
.13 
.13 
.13 
10 

.792662 
.794701 
.796731 
.798752 
.800763 
.802765 

34.  15 
33.99 
33.83 
33.63 
33.52 
33.37 

.207338 
.205299 
.203269 
.201248 
.199237 
.197235 

27 
26 
25 
24 
23 
22 

39 

.803876 

oo.uo 
32.93 

.999113 

.Id 

.13 

.804758 

33.22 
33.07 

.195242 

21 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

8.805852 
.807819 
.809777 
.811726 
.813667 
.815599 
.817522 
.819436 
.821343 
.823240 

32.78 
32.63 
32.49 
32.34 
32.20 
32.05 
31.91 
31.77 
31.63 
31.49 

9.999110 
.999102 
.999094 
.999086 
.999077 
.999069 
.999061 
.999053 
.999044 
.999036 

.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 

8.806742 
.808717 
.810683 
.812641 
.814589 
.816529 
.818461 
.820384 
.822298 
.824205 

32.92 
32.77 
32.62 
32.48 
32.33 
32.19 
32.05 
31.91 
31.77 
31.63 

1.193258 
.191283 
.189317 
.187359 
.185411 
.183471 
.181539 
.179616 
.177702 
.175795 

20 

19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

8.825130 
.827011 
.828884 
.830749 
.832607 
.834456 
.836297 
.838130 
.839956 
.841774 
.843585 

31.36 
31.22 
31.08 
30.95 
30.82 
30.69 
3056 

30.'30 
30.17 

9.999027 
.999019 
.999010 
.999002 
.993993 
.993984 
.993976 
.998967 
.993958 
.998950 
.998941 

.14 
.14 
.14 
.14 
.14 
.14 
.15 
.15 
.15 
.15 

8.826103 
.827992 
.829374 
831748 
.833613 
.835471 
.837321 
.839163 
.840998 
.842825 
.844644 

3J  50 

31.23 
31.09 
30.96 
30.83 
30.70 
30.57 
30.45 
30.32 

1.173897 

.172008 
.170126 
.168252 
.166387 
.164529 
.162679 
.160837 
.159002 
.157175 
.155356 

10 
9 
8 
7 
6 
5 
4 
3 
2 

0 

M. 

Cosine. 

D.I" 

Slue. 

D.  1". 

Cotang. 

D.l". 

Tang. 

M. 

93" 


234 


TABLE    II.       LOGARITHMIC    SINES, 


M 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D  1". 

Cotang. 

M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

8.843585 
.845387 
.847183 
.848971 
.850751 
.852525 
.854291 
.856049 
.857801 
.859546 

30.05 
2992 
29.  SO 
2963 
29.55 
29.43 
29.31 
29.19 
29.08 
28.96 

9.998941 
.998932 

.998'.t23 
.998914 
.998905 
993896 
.998887 
.998873 
.998869 
.99886C 

.15 
.15 
.15 
.15 
.15 
.15 
'5 
.15 
.15 
.15 

8.844644 
.846455 
.848260 
.850057 
.851846 
.853628 
.855403 
.857171 
.858932 
860686 

30.20 
30.07 
29.95 
29.83 
29.70 
29.58 
29.46 
29.35 
29.23 
29  11 

1.155356 
.153545 
.151740 
.149943 
.148154 
.146372 
.144597 
.142829 
.141068 
.139314 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

8.861283 
.863014 
.864738 
.866455 
.863165 
.869863 
.871565 
.873255 
.874933 
.876615 

23.84 
28  ~3 
23.61 
23.50 
23.39 
28.2* 
28.17 
28.06 
27.95 
27.84 

9.998851 
.998841 

.998832 
.998823 
.998813 
.998804 
998795 
.998785 

'.  998766 

.15 
.15 
.15 
.16 
.16 
16 
.16 
16 
.16 
.16 

8.862433 
.864173 
.865906 
.867632 
.869351 
.871064 
.872770 
.874469 
.876162 
.877849 

29.00 
128.88 
28.77 
28.66 
28.55 
28.43 
28.32 
28.22 
28.11 
28.00 

1.137567 
.135827 
.134094 
.132368 
.130649 
.128936 
127230 
.125531 
.123838 
.122151 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

8.878285 
.879949 
.881607 
.883253 
884903 
.886542 
883174 
.889801 
891421 
.893035 

27.73 
2763 
27  52 
2742 
2731 
27.21 
27.11 
27.  00 
26.90 
26.80 

9.998757 
.998747 
.998733 
.99*728 
99^719 
91)8703 
9i)>699 
99*689 
.99^79 
.998669 

.16 
.16 
.16 
.6 
.16 
.16 
.16 
.16 
.16 
.17 

8.879529 
.881202 
.882869 
.884530 
.886185 
.887833 
.889476 
.891112 
892742 
.894366 

27.89 
2779 
27.68 
27.58 
2747 
27.37 
27.27 
27  17 
2707 
26.97 

1.120471 
.118798 
.117131 
.115470 
.113815 
.112167 
.110524 
.108888 
.107258 
.105634 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 

8.894643 
.896246 

26.70 
26.60 

9.998659 
.998649 

.17 
17 

8.895984 
.897596 

26.87 
2677 

1.104016 
.102404 

30 
29 

33 

34 
35 
36 
37 
38 
39 

.899432 
.901017 
.902596 
.904169 
.905736 
.907297 
.908853 

26.51 
26.41 
26.31 
2622 
26.12 
26.03 
25.93 
25.84 

.998639 
.998629 
998619 
.998609 
.998599 
998589 
998578 
993568 

.17 
.17 
.17 
.17 
.17 
.17 
.17 
.17 

.899203 
.900803 
.902398 
903937 
.905570 
.907147 
.908719 
.910235 

26.67 
26.58 
2648 
26.39 
26.29 
26.20 
26.10 
26.01 

.100797 
.099197 
.097602 
.096013 
.094430 
.092853 
.091281 
.089715 

27 
26 
25 
24 
23 
22 
21 

40 
41 
42 

8.910404 
.911949 
.913488 

25.75 
2566 

9.998558 
.993548 
.998537 

.17 
.17 

8.911846 
.913401 
.914951 

25.92 
25.83 

1.088154 
.086599 
.085049 

20 
19 
IS 

43 

44 
45 

46 
47 

43 
49 

.915022 
.916550 
.918073 
.919191 
.921103 
.922610 
.924112 

25.47 
25.33 
25.29 
25.21 
25.12 
25.03 
24.94 

998527 
.998516 
.998506 
.998495 
.9984^5 
.998474 
.998464 

.17 
.17 
.18 
.18 
.18 
.13 
.18 

.916495 
.913034 
.919568 
.921096 
.922619 
.924136 
.925649 

25.65 
25.56 
25.47 
25.38 
25.29 
25.21 
25.12 

.083505 
.081966 
.080432 
.078904 
.077381 
.075864 
.074351 

17 
16 
15 
14 
13 
12 
11 

50 
51 

8.925609 
.927100 

2486 

9.993453 
.998442 

.18 

8.927156 
.923658 

25.04 

1.072844 
.071342 

10 
9 

52 
53 
54 

.928587 
.930068 

2469 
24.60 

.998431 
.998421 
.998410 

.18 
18 

.930155 
.931647 
.933134 

24.87 
2473 

.069845 
.068353 
.066866 

8 
7 
6 

55 
56 
57 
58 
59 
60 

.933015 
.934481 
.935942 
.937398 
.938350 
.940296 

24.43 
24.35 
24.27 
24.19 
24.11 

.998399 
.998388 
.998377 
.998366 
.998355 
.998344 

.18 
.18 
.18 
.18 
.18 

.934616 
.936093 
.937565 
.939032 
.940494 
.941952 

24.62 
24.53 
24.45 
24.37 
24.29 

.065384 
.063907 
.062435 
.U60968 
.059506 
.058048 

5 
4 
3 
2 

1 
0 

M. 

Ooelue. 

D.I'. 

Sine. 

D.  1". 

Cotang.   D.  1'  . 

T«ug. 

M 

94° 


85° 


COSINES,  TANGENTS,  AND  COTANGENTS.         235 

BO                                               17*0 

11 

Sine. 

D.P. 

Cosine. 

D.  1". 

Tang. 

D.  I'. 

Gotang. 

M. 

1 
2 

8.940296 
.941738 
.943174 

24.03 
23.95 

9.993344 
.998333 
.998322 

.18 
.19 

8.941952 
.943404 
.944352 

24.21 
24.13 

1.058048 
.056596 
.055148 

60 
59 

58 

3 
4 
5 

.944606 
.946034 
.947456 

23.79 
2371 

.998311 
.998300 
.998289 

.19 
.19 

.916295 
.947734 
.949168 

23.97 
23.90 

.053705 
.052266 
.050832 

57 
66 
65 

6 
7 

6 
9 

.948874 
.950287 
.951696 
.953100 

23.55 
23.48 
23.40 
23.32 

.998277 
.998266 
.998255 
.998243 

.19 
.19 
.19 
.19 

.950597 
.952021 
953441 
.954856 

23.74 
23.67 
23.59 
23.51 

.049403 
.047979 
.046559 
.045144 

54 
63 
52 
61 

10 
11 
12 
13 
14 
15 

8.954499 
.955394 
.957284 
.958670 
.960052 
.961429 

23.25 
23.17 
23.10 
23.02 
22.95 

9.998232 
.998220 
.993209 
.993197 
.998186 
.998174 

.19 
.19 
.19 
.19 
.19 

8.956267 
.957674 
.959075 
.960473 
.961866 
.963255 

23.44 
23.36 
23.29 
23.22 
23.14 

1.043733 
.042326 
.040925 
.039527 
.038134 
.036745 

50 
49 
48 
47 
46 
45 

16 
17 
18 
19 

962301 
.964170 
.965534 
.966393 

22.81 
22.73 
22.66 
22.59 

.993163 
.998151 
.993139 
.998128 

.19 
.20 
.20 
.20 

.964639 
.966019 
.967394 
.968766 

23.00 
22.93 
22.86 
22.79 

.035361 
.033981 
.032606 
.031234 

44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 

8.963249 
.969600 
.970947 
.972239 
.973623 
.9749f,2 
.976293 
.977619 
.978941 
.980259 

22.52 
22.45 
22.38 
22.31 
22.24 
22.17 
22.10 
22,03 
21.97 
21.90 

9.998116 

.998104 
.993092 
.993080 
.998068 
.998056 
.998044 
.993032 
.998020 
.998008 

.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 

8.970133 
.971496 
.972855 
.974209 
.975560 
.976906 
.978243 
.979586 
.930921 
.932251 

22.72 
22.65 
22.58 
22.51 
22.44 
22.37 
22.30 
22.24 
22.17 
22.10 

1.029867 
.023504 
.027145 
.025791 
.024440 
.023(194 
.021752 
.020414 
.019079 
.017749 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 

8.981573 
.93S333 
.984189 
.985491 

2183 
21.77 
21.70 

9.997996 
.997984 
.997972 
.997959 

.20 

.20 

.20 

8.983577 
.934899 
986217 
.987532 

22.04 
21.97 
21.91 

1.016423 
.015101 
.013783 
.012468 

30  | 
29 
28 
27 

34 
35 
36 
37 

38 
39 

.986789 
.988033 
.989374 
.990660 
.991943 
.993222 

21.57 
21.51 
21.44 
21.38 
21.31 
21.25 

.997947 
.997935 
.997922 
.997910 
.997897 
.997885 

.21 
.21 
.21 
.21 
.21 
.21 

.988842 
.990149 
.991451 
.992750 
.994045 
.995337 

21.78 
21.71 
21.65 
21.59 
21.52 
21.46 

.011158 
.009351 
.008549 
.007250 
.005955 
.004663 

26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

8.994497 
.995768 
.997036 
.998299 
.999560 
9.000316 
.002069 
.003318 
.004563 
.005805 

21.19 
21.12 
21.06 
21.00 
20.94 
20.88 
20.82 
20.76 
20.70 
20.64 

9.997872 
.997860 
.997847 
.997835 
.997822 
.997809 
.997797 
.997734 
.997771 
.997758 

21 
.21 
.21 
.21 
.21 
.21 
.21 
.21 
.21 
.21 

8.996624 
.997903 
.999138 
9.000465 
.001738 
.003007 
.004272 
.005534 
.006792 
.003047 

21.40 
21.34 
21.27 
21.21 
21.15 
21.09 
21.03 
20.97 
20.91 
20.85 

1.003376 
.002092 
.000812 
0.999535 
.993262 
.996993 
.995728 
.994466 
.993208 
.991953 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.007044 
.003278 
.009510 
.010737 
.011962 
.013182 
.014400 
.015613 
.016824 
.018031 
.019235 

20.58 
20.52 
20.46 
20.40 
20.35 
20.29 
2023 
20.17 
20.12 
20.06 

9.997745 
.997732 
.997719 
.997706 
.997693 
.997690 
.997667 
.997654 
.997641 
.997623 
.997614 

.22 
.22 

.22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 

9.009298 
.010546 
.011790 
013031 
.014263 
.015502 
.016732 
.017959 
.019183 
.020403 
.021620 

20.80 
20.74 
20.68 
2062 
20.56 
20.51 
20.45 
20.39 
20.34 
20.28 

0.990702 
.989454 
.988210 
.986969 
.985732 
.984498 
.983*68 
.982041 
.9*0817 
.979597 
.978380 

10 
9 
8 
7 
6 
6 
4 
3 
2 

0 

M. 

Oosina  |  D.  1". 

Sine. 

D.  1". 

Gotang. 

D.  1". 

Tang. 

M. 

236 


TABLE    II.       LOGARITHMIC    SINES, 


M. 

Sine. 

D.  1". 

Cosine. 

D  1". 

Tang. 

D.  1". 

Cotang. 

M. 

1 

2 

9.019235 
.020435 
.021632 

20.00 
19.95 

9.997614 
.997601 
.997588 

.22 
.22 

9.021620 
.022834 
.024044 

20.23 
20.17 
on  i  o 

0.978380 
.977166 
.975956 

lo" 

59 

58 

3 
4 

6 
6 
7 
8 
9 

.022325 
.024016 
.025203 
.026386 
.027567 
.028744 
.029918 

19.89 
19.84 
19.78 
19.73 
19.67 
19.62 
19.57 
19.51 

.997574 
.997561 
.997547 
.997534 
.997520 
.997507 
.997493 

!22 
.22 
22 
.23 
.23 
23 
23 

.025251 
.026455 
.027655 
.028852 
.030046 
.031237 
.032425 

SRf.l* 

20.06 
20.01 
19.95 
19.90 
19.85 
19.79 
19.74 

.974749 
.973515 
.972345 
.971148 
.969954 
.968763 
.967576 

67 
56 
65 
64 
53 
52 
61 

10 
11 
12 
13 
14 
15 

9.031089 
032257 
.033421 
.034582 
.035741 
.036896 

19.46 
19.41 
19.36 
19.30 
19.25 
ici  on 

9.997480 
.997466 
.997452 
.997439 
.997425 
.997411 

23 
23 
.23 
23 
.23 
oo 

9.033609 
.034791 
.C35969 
.037144 
.038316 
.039485 

19.69 
19.64 
19.58 
1953 
19.48 

1Q  Al 

0.966391 
.965209 
.964031 
.962856 
.961684 
.960515 

60 
49 
48 
47 
46 
45 

16 
17 
13 
19 

.038048 
.039197 
.040342 
.041485 

Iw.SHI 

19.15 
19.10 
19.05 
19.00 

.997397 
.997333 
.997369 
.997355 

JM 

.23 
.23 
.23 
.23 

.040651 
.041813 
.042973 
.044130 

llf.M 

19.38 
19.33 
19.28 
19.23 

.959349 
.958187 
.957027 
.955870 

44 
43 
42 
41 

20 
21 
22 
23 

9.042625 
.043762 
.044895 
.046026 

18.95 
18.90 

18.85 

9.997341 
.997327 
.997313 
.997299 

.23 
.23 
.24 

9.045284 
.046434 
.047582 
.048727 

19.18 
19.13 
19.08 

0.954716 
.953566 
.952418 
.951273 

40 
39 
38 
37 

24 
25 
26 

.047154 
.048279 
.049400 

18.80 
18.75 
18.70 

.997285 
.997271 
.997257 

.24 
.24 
.24 

.049869 
051008 
.052144 

19.03 
18.98 
18.93 

.950131 
.948992 
.947856 

36 
35 
34 

27 
28 
29 

.050519 
.05M35 
.052749 

18.65 
18.60 
18.65 
18.50 

.997242 
.997228 
.997214 

.24 

.24 
.24 
.24 

.053277 
.054407 
.055535 

18.89" 
18.84 
1879 
18.74 

.946723 
.945593 
.944465 

33 
32 
31 

10 
31 

32 
33 
34 
36 
36 
37 
38 
39 

9.053859 
.054966 
.056071 
.057172 
.058271 
.059367 
.06(M60 
.061551 
.062639 
.063724 

18.46 
1841 
18.36 
18.31 
1827 
18.22 
18.17 
18.13 
18.08 
18.04 

9.997199 
.997186 
.997170 
.997156 
.997141 
.997127 
.997112 
.997098 
.997(183 
.997068 

.24 
.24 
.24 
.24 
.24 
.24 
.24 
.24 
.25 
.25 

9.066659 
.057781 
.058900 
.060016 
.061130 
.062240 
.063348 
.064453 
.065556 
.066655 

18.70 
18.65 
18.60 
1856 
18.51 
1846 
18.42 
18.37 
18.33 
18.28 

0.943341 
.942219 
.941100 
.939984 
.938870 
.937760 
.936652 
.935547 
.934444 
.933345 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 

9.064806 
.065885 
.066962 
.06S036 
.069107 
.070176 
.071242 
.072306 
.073366 

17.99 
17.95 
17.90 
17.86 
17.81 
17.77 
17.72 
17.68 

9.997053 
.997039 
.997024 
.997009 
.996994 
.996979 
.996964 
.996949 
.996934 

.25 
.25 
.25 
.25 
.25 
.25 
.25 
.25 

9.067752 
.068846 
.069938 
.071027 
.0721  13 
.073197 
.074278 
.075356 
.076432 

18.24 
18.19 
18.15 
18.10 
18.06 
18.02 
17.97 
17.93 

0.932248 
.931154 
.930(62 
.928973 
.927887 
.9268(0 
.925722 
.924644 
.923568 

20 
19 
18 
17 
16 
15 
14 
13 
12 

49 

.074424 

17.64 
17.59 

.996919 

.25 
.25 

.077505 

17.89 
17.84 

.922495 

11 

50 
51 

3.075480 
.076533 

17.55 

9.996904 
.996889 

.25 

9.078576 
.079644 

17.80 
.»  .*!• 

0.921424 
.920356 

10 

9 

62 
53 
54 
55 
66 
57 
58 

.077583 
.078631 
.079676 
.080719 
.081759 
.082797 
.083832 

17.51 
17.46 
17.42 
17.38 
17.34 
17.29 
17.25 

.996874 
.996858 
.996843 
.996828 
.996812 
.996797 
.996782 

.25 
.25 
.25 
.26 
.26 
.26 
.26 

.080710 
.081773 
.082833 
.083891 
.084947 
.086000 
.087050 

17.76 
17.72 
17.67 
17.63 
17.59 
17.55 
17.51 

.919290 
.918227 
.917167 
.916109 
.915053 
.914000 
.912950 

8 
7 
6 
6 
4 
3 
2 

59 
60 

.084864 
.085894 

17.21 
17.17 

.996766 
.996751 

.26 
.26 

.088098 
.089144 

17.47 
17.43 

.911902 
.910S56 

0 

HI. 

Codne. 

D.  1". 

Sine 

D.I. 

Cotang. 

D.I". 

Tang. 

M 

83° 


COSINES,    TANGENTS,    AND   COTANGENTS. 


237 


70 


M. 

Blue. 

D.  1". 

Cosine. 

D.1*. 

Tang. 

D.  i«. 

Coiaiig 

U. 

0 
1 
2 
3 

4 
5 
8 
7 
8 
9 

9.085894 
.086922 
.087947 
.088970 
.089990 
.091003 
.092024 
.093037 
.094047 
.095056 

17.13 

17.09 
17.05 
17.00 
16.96 
16.92 
16.88 
16.84 
16.80 
16.76 

9.996751 
.996735 
.996720 
.996704 
996688 
996673 
996657 
.996641 
.996625 
.996610 

.26 
.26 
.26 
.26 
.26 
.26 
.26 
.26 
.26 
.26 

9.089144 
.090187 
.091228 
.092266 
.093302 
094336 
.095367 
096395 
.097422 
.098446 

17.39 
17.35 
17.31 
17.27 
17.23 
17.19 
17.15 
17.11 
17.07 
17.03 

0.910856 
.909313 
.908772 
.907734 
.9UK93 
.905664 
.9046:53 
.903605 
.902578 
.901554 

80 
69 
68 
67 
56 
65 
51 
63 
52 
61 

10 
11 
12 
13 
14 
15 
16 
17 

9.096062 
.097065 
.093066 
.099065 
100062 
.101056 
102048 
.103037 

16.73 
16.69 
16.65 
16.61 
16.57 
16.53 
16.49 

9.996594 
996578 
996562 
996546 
996530 
996514 
996498 
996482 

.27 
.27 
.27 
.27 
.27 
.27 
.27 

9.099468 
.100487 
•101504 
102519 
103532 
.104542 
105550 
.106556 

16.99 
16.95 
16.91 
16.88 
16.84 
16.80 
16.76 

0.900532 
.899513 
.893496 
.897481 
.896468 
.895458 
.894450 
.893444 

60 
49 
48 
47 
46 
46 
44 
43 

18 

.104025 

99G4C5 

27 

.107559 

16  69 

.892441 

42 

19 

.105010 

16.38 

.996449 

.27 

.108560 

1&6S 

.891440 

41 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

9.105992 
106973 
107951 
.108927 
.109901 
.110873 
.111842 
.112809 
.113774 
.114737 

16.34 
16.30 
16.27 
16.23 
16.19 
16.16 
16.12 
16.08 
16.05 
16.01 

9.996433 
996417 

.996400 

9963GS 
.996351 
996335 
996318 
.996302 
996285 

.27 

.27 
.27 
.27 
.27 
.27 
.23 
.28 
.28 
.28 

9.109559 
.110556 
.111551 
.112543 
.113533 
.114521 
.115507 
.116491 
.117472 
.118452 

16.61 
16.58 
16.54 
16.50 
16.47 
16.43 
16.39 
16.36 
16.32 
16.29 

0.890441 
.889444 

.888449 
.887457 
.886467 
.885479 
.884493 
.883509 
.882528 
.881548 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.115698 
.116656 
.117613 
.118567 
.119519 
.120469 
.121417 
.122362 
.123306 
.124248 

T5.98 
1594 
15.90 
15.87 
15.83 
15.80 
15.76 
15.73 
15.69 
15.66 

9.996269 
.996252 
.996235 
.996219 
.996202 
996185 
996168 
996151 
996134 
.996117 

.28 
.28 
.28 
.28 
.23 
.23 
.23 
28 
.28 
.28 

9.119429 
.120404 
.121377 
.122.-M8 
.123317 
124284 
125249 
126211 
127172 
128130 

16.25 
16.22 
16.18 
16.15 
16.11 
16.03 
16.04 
16.01 
15.93 
15.94 

0.880571 
.879596 

.878623 
.877652 
.876683 
.875716 
.874751 
.873789 
.872823 
.871870 

30 
29 
28 
27 
26 
26 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
48 
47 
48 
49 

9.125187 
.126125 
.127060 
.127993 
.128925 
.129854 
.130781 
.131706 
.132630 
.133551 

15.62 
15.59 
15.56 
15.52 
15.49 
15.45 
15.42 
15.39 
15.35 
15.32 

9.996100 
996083 
996066 
996049 
.996032 
.996015 
.995998 
.995980 
995963 
995946 

.28 

.28 
.28 
.29 
.29 
.29 
.29 
.29 
.29 
.29 

9.129087 
130041 
.130994 
.131944 
.132893 
.133339 
.134784 
.135726 
.136667 
.137605 

15.91 
15.87 
15.84 
15.81 
16.77 
15.74 
16.71 
15.68 
15.64 
16.61 

0.870913 
.869959 
.869006 
.868056 
.867107 
.866161 
.8652t6 
.864274 
.863333 
.862395 

20 
19 
18 
17 
16 
16 
14 
13 
12 
11 

60 
51 
52 

9.134470 
.135337 
.136303 

15.29 
15.26 

9.995928 
.995911 
995894 

.29 
.29 

9.138542 
139476 
140409 

15.58 
15.55 

IE  C1 

0.861453 

.860524 
.859591 

10 
9 

8 

53 
64 
65 
56 
57 

137216 
.138128 
.139037 
.139944 

.140S50 

15.19 
1516 
15.13 
15.09 

995876 
995.359 
995841 
995,323 
995806 

.29 
.29 
.29 
.29 

141340 
142269 
.143196 
144121 
.14S044 

15.43 
15.45 
15.42 
15.39 

.858660 
.857731 
.856804 
.855879 
.854956 

7 
6 

6 
4 
3 

68 
59 

141754 
142655 

15.03 

995788 
995771 

.29 

.145966 
146885 

15.32 

.854034 
.853115 

2 

60 

.143555 

.995753 

.147803 

.852197 

0 

M. 

Cosine. 

D.  i». 

Blue. 

D.  1". 

Ootaug. 

D.  1'  . 

•Ring. 

M 

238 


TABLE    II.       LOGARITHMIC    SINES, 


M. 

Sine 

D.  1". 

Ccelne. 

D.I' 

Twig. 

D.  1". 

Cotang 

M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

9  143555 
.144453 
.145349 
.146243 
.147136 
.148026 
.148915 
.149802 
.150686 
.151569 

14.97 
14.93 
14.90 
14.87 
14.84 
14.81 
14.78 
14.75 
14.72 
14.69 

9.995753 
.995735 
.995717 
.995699 
995681 
.995664 
.995646 
.995628 
.995610 
.995591 

.30 
.30 
.30 
.30 
.30 
.30 
.30 
.30 
.30 
.30 

9.  147803 
.148718 
.149632 
.150544 
.151454 
152363 
.153269 
154174 
155077 
.155978 

15.26 
15.23 
15.20 
15.17 
15.14 
15.11 
15.08 
15.05 
15.02 
14.99 

0.852197 
.851282 
.850368 
.849456 
.848546 
.847637 
.846731 
.845826 
.844923 
.844022 

60 
59 
58 
67 
56 
55 
54 
53 
52 
61 

10 
11 

9.152451 
.153330 

14.66 

9.995573 
.995555 

.30 

9.156877 
.157775 

14.96 

0.843123 
.842225 

50 

49 

12 
13 
14 
15 
16 
17 
18 
19 

154208 
155(183 
155957 
.156830 
.157700 
.158569 
.159435 
.160301 

14.60 
14.57 
14.54 
14.51 
14.48 
14.45 
14.42 
14.39 

.995537 
.995519 
.995501 
.995482 
.995464 
.995446 
.995427 
.995409 

.30 
.30 
.30 
.30 
31 
.31 
.31 
.31 
.31 

.158671 
.159565 
.160457 
.161347 
.162236 
.163123 
.164008 
.164892 

14.93 
1490 
14.87 
14.84 
14.81 
14.78 
14.75 
14.73 
14.70 

.841329 
.840435 
.839543 
.838653 
.837764 
.836877 
835992 
.835108 

48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.161164 
.162025 
.162885 
.163743 
.164600 
.165454 
.166307 
.167159 
168008 
.168856 

14.36 
14.33 
14.30 
14.27 
14.24 
14.22 
14.19 
14.16 
14.13 
14.10 

9.995390 
.995372 
.995353 
.995334 
.995316 
.995297 
.995278 
995260 
.995241 
.995222 

.31 
.31 
31 
.31 
.31 
.31 
.31 
.31 
.31 
.31 

9.165774 
.166654 
.167532 
.168409 
.169284 
.170157 
.171029 
.171899 
.172767 
.173634 

14.67 
14.64 
14.61 
14.58 
14.56 
14.53 
14.50 
14.47 
14.44 
14.42 

0.834226 
.833346 
832468 
.831591 
.830716 
.829843 
.828971 
.828101 
.827233 
.826366 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 

9.169702 
.170547 

14.07 

9.995203 
.995184 

.31 

9.174499 
.175362 

14.39 

0.825501 
.824688 

30 
29 

32 
33 
34 
35 

36 
37 
38 
39 

.171389 
.172230 
.173070 
.173908 
.174744 
.175578 
176411 
.177242 

14.02 
13.99 
13.96 
13.94 
13.91 
13.88 
13.85 
13.83 

.995165 
.995146 
.995127 
.995108 
.995089 
.995070 
.995051 
.995032 

.32 
.32 
.32 
.32 
.32 
.32 
32 
.32 
.32 

.176224 

.177084 
.177942 
.178799 
.179655 
.180508 
.181360 
.182211 

14.33 
14.31 
14.28 
14.25 
14.23 
14.20 
14.17 
14.15 

.823776 
.822916 
.822058 
.821201 
.820345 
.819492 
.818640 
.81778? 

23 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 

9.178072 
.178900 
.179726 
.180551 

13.80 
13.77 
13.75 

9.995013 
.994993 
.994974 
.994955 

.32 
.32 
.32 

9.183059 
.183907 
.184752 
.185597 

14.12 
14.09 
14.07 

0.816941 
.816093 
.815248 
.814403 

20 
19 
18 
17 

44 
'15 

46 
47 

48 
49 

.181374 
.182196 
.183016 
.183834 
.184651 
.185466 

13.69 
13.67 
13.64 
13.61 
13.59 
13.56 

.994935 
.994916 
.994896 
.994877 
.994857 
.994838 

.32 
.32 
.33 
.33 
.33 
.33 

.186439 
.187280 
.188120 
.188958 
.189794 
.190629 

14.02 
13.99 
13.97 
13.94 
13.91 
13.89 

.813561 
.812720 
811880 
.811042 
.810206 
.809371 

16 
16 
14 
13 
12 
11 

60 
51 
52 
53 
54 
55 
66 
67 
58 
59 

9.186280 
187092 
.187903 
.188712 
.189519 
.190325 
.191130 
.191933 
.192734 
.193534 

1364 
13.51 
13.48 
13.46 
13.4J 
13.41 
13.38 
13.36 
13.33 

9.994818 
.994798 
.994779 
.994759 
.994739 
.994720 
994700 
.994680 
994660 
994640 

.33 
.33 
.33 
.33 
.33 
.33 
.33 
.33 
.33 

9.191462 
.192294 
.193124 
.193953 
.194780 
.195606 
.196430 
.197253 
.198074 
.198894 

13.86 
13.84 
13.81 
13.79 
13.76 
13.74 
13.71 
13.69 
13.66 

0.808538 
'  .807706 
.806876 
.806047 
.805220 
.804394 
.803570 
.802747 
.801926 
.801106 

10 
9 
8 
7 
6 
6 
4 
3 
2 

60 

.194332 

.994620 

199713 

.800287 

0 

M. 

Conine. 

D.  1". 

Sloe. 

D.  1". 

Cotang. 

D.  1". 

Taug. 

M. 

980 


COSINES,  TANGENTS,  AND  COTANGENTS.        239 

90                                               1TO" 

M. 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tung. 

D.  i»: 

Cotang. 

M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

9.194332 
.195129 
.195925 
.196719 
.197511 
.198302 
.199091 
.199379 
.200666 
.201451 

13.28 
13.26 
13.23 
13.21 
13.18 
1316 
13.13 
13.1 
13.08 
13.06 

97994620" 
.994600 
.994580 
.994560 
.994541) 
.994519 
.994499 
.994479 
.994459 
.994438 

.33 
.33 
.34 
.34 
.34 
.34 
.34 
.34 
.34 
.34 

9.199713 

.200529 
.201345 
.202159 
.202971 
.2113732 
.204592 
.205400 
.206207 
.207013 

J3.62 
13.59 
13.57 
13.54 
13.52 
13.49 
13.47 
13.45 
13.42 
13.40 

0.800287 
.799471 
.798655 
.797841 
.797029 
.796218 
.795403 
.794600 
.793793 
.792987 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 

9.202234 
.203017 
.203797 

13.04 
13.01 

1O  Oil 

P  9944  18 
994398 
.994377 

.34 
.34 
04 

9.207817 
.208619 
.209420 

13.38 
13.35 

10  qq 

0.792183 
.791331 

.790530 

50 
49 

43 

13 
14 
15 
16 
17 
18 
19 

204577 
205354 
206131 
206906 
207679 
203452 
209222 

IX.  M) 

1296 
12.94 
12.92 
i2.89 
12.87 
12.85 
12.32 

.994357 
.994336 
994316 
.994295 
.994274 
.9942.54 
.994233 

.<>* 
.34 
.34 
.34 
.34 
.34 
.35 
.36 

.210220 
.211013 
.211815 
.212611 
.213405 
.214193 
.214989 

ia.oo 

13.31 
13.23 
13.26 
13.24 
13.21 
13.19 
13.17 

.789780 
.788982 
.788185 
.787.3S9 
.78659.1 
.785802 
.785011 

47 
46 

45 
44 
43 
42 
41 

20 
21 
22 
23 
24 

9.209992 
210760 
211526 
.212291 
.213056 

12.80 
12.78 
1275 
1273 

9.994212 
.994191 
.994171 
.994150 
.994129 

.35 
.35 
.35 
.35 

oc 

9.215780 
216568 
.217356 
.218142 
.218926 

13.15 
13.12 
13.10 
13.08 
i  o  nc 

0.784220 
.783432 

.782644 
.781858 
.781074 

40 
39 
38 
37 
36 

25 
26 
27 
28 
29 

21381S 
214579 
215338 
216097 
216854 

12.71 
1268 
12.66 
12.64 
12.62 
12.59 

.994108 
.994037 
.994066 
.994045 
.994024 

•ID 

.35 
.35 
.35 
.35 
.35 

.219710 

.220492 
221272 
2220.12 
222330 

lo.UO 

13.03 
13.01 
12.99 
12.97 
12.95 

.780290 
.779508 
.778728 
.777948 
.777170 

35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.217609 
218363 
819116 

219368 
220618 
221367 
222115 
222861 
.223606 
.224349 

12.57 
12.55 
12.53 
12.50 
12.48 
12.46 
12.44 
12.42 
12.39 
12.37 

9.994003 
9939S2 
993960 
993939 
.993918 
.993397 
.993375 
.993354 
.993332 
.993811 

.35 
.3.1 
.35 
.35 
.36 
.36 
.36 
.36 
.36 
.36 

9.223607 
224332 
225156 
225929 
226700 
.227471 
223239 
.229007 
.229773 
.230539 

12.92 
12.90 
12.88 
12.86 
12.84 
12.82 
12.79 
12.77 
12.75 
12.73 

0.776393 
.775618 
.774844 
.774071 
.773300 
.772529 
.771761 
.770993 
.770227 
.769461 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 

9.225092 
225833 
.226573 
.227311 

12.35 
12.33 
12.31 

9.993789 
993768 
993746 
.993725 

.36 
.36 
.36 

9.231302 
232065 
232326 
233586 

12.71 
12.69 
12.67 

0.768698 
.767935 
.767174 
.766414 

20 
19 
18 
17 

44 
45 
46 
47 

.228048 
.228784 
.229518 
.230252 

12.29 
12.26 
12.24 
12.22 

.993703 
.993681 
.993660 
993633 

.36 
.36 
.36 
.36 

234345 
235103 
235359 
.236614 

12  65 
12.63 
12.60 
12.58 

.765655 
.764397 
.764141 
.763386 

16 
15 
14 
13 

48 
49 

.230934 
.231715 

12.20 
12.18 
12.16 

.993616 

.36 
.36 
.36 

.237363 
233120 

12.56 
12.54 
12.52 

.762632 
.761880 

12 
11 

50 

9.232444 

9.993572 

9.238872 

0.761  128 

10 

51 
52 
53 
54 
55 
56 
57 

233172 
233399 
234625 
235349 
.236073 
.236795 
.237515 

12.14 
12.12 
12.10 
12.07 
12.05 
12.03 
12.01 

993550 
993528 
.9935(16 
.993484 
.993462 
.993440 
.993413 

.37 
.37 
.37 
.37 
.37 
.37 
37 

.239622 
240371 
241118 
.241865 
242610 
.243354 
.244097 

12.50 
12.48 
12.46 
12.44 
12.42 
12.40 
12.33 

.760378 
.759629 
.758882 
.758135 
.757390 
.756646 
.755903 

9 
8 
7 
6 
5 
4 
3 

58 
59 

.2382.'6 
.233953 

11.99 
11.97 

993396 
.993374 

.37 

•£ 

244339 
.245579 

12.36 
12.34 

.755161 
.754421 

2 

I 

60 

.23967** 

11.95 

.993351 

.37 

.216319 

12.32 

.753681 

0 

M. 

Cosine.   D.  1". 

Sine. 

D.1". 

Cotaiig. 

D.  1" 

Tang. 

M. 

99° 


800 


240 

HP 


TABLE    II.      LOGARITHMIC    SINES, 


».239670 
.240386 
.241101 
.941814 
.842686 
.243237 
.243947 
.244656 
.245363 


9.246775 
.247478 
.248181 


.261677 
.252373 
.253067 

9.253761 
.254463 
.255144 
.855834 


.257211 


.259268 
.259951 


.261314 


.264027 


M.      Cosine. 
1000 


.265377 

.266051 


9.267395 


.268734 


.270069 

.270735 
.271400 
.272064 
.272726 


9.2T4049 
.274708 
.275367 
.276025 
.276681 
.277337 
.277991 
.278645 


.279948 


D.  1".      Cosine. 


11.93 
11.91 
11.89 
11.87 
11.85 
11.83 
11.81 
11.79 
11.77 
11.75 

11.73 
11.71 
11.69 
11.67 
11.66 
11.63 
11.61 
11.59 
11.58 
11.56 

11.54 
11.52 
11.60 
11.48 
11.46 
11.44 
11.42 
11  41 
11.39 
11.37 

11.35 
11.33 
11.31 
11.30 
11.28 
11.26 
11.24 
11.22 
11.20 
11.19 

11.17 
11.16 
11.13 
11.12 
11.10 
11.08 
11.06 
11.05 
11.03 
11.01 

10.99 
10.98 
10.96 
10.94 
10.92 
10.91 
10.89 
10.87 
10.86 
10.84 

D.  1". 


9.993351 
.993329 
.993307 
.993284 


993240 
.993217 
.993195 
.993172 
.993149 
9.993127 
.993104 
.993081 
.993059 
.993036 
.993013 


.992967 


.992375 


.992783 


.992736 

.992713 


.992572 


.992335 


.992287 
.992263 


.992214 

9.992190 
.992166 
.992142 
.992118 

.992093 


.992044 
.992020 
.991996 
.991971 
.991947 


D.  1". 


.39 


.40 
.40 

.40 

.40 
.40 
.40 
.40 
.40 
.40 
.40 
.40 
.40 

.40 
.40 
.40 
.41 
.41 
.41 
.41 
.41 
.41 
.41 

D.  1". 


9.246319 
.247057 
.247794 
.248530 
.249264 
.249998 
.250730 
.251461 
.252191 


9.253648 
.254374 
.255100 
.255824 
.256547 
.257269 
.257990 
.258710 


.260146 

9.260863 
.261578 
.262292 
.263005 
.263717 
.264428 
.265138 
.265847 
.266555 
.267261 

9.267967 
.268671 
.269375 
.270077 
.270779 
.271479 
.272178 
.272876 
.273573 


9.274964 
.275658 
.276351 
.277043 
.277734 
.278424 
.279113 
.279801 
.230488 
.281174 


.282542 
.283225 
.283907 


.285947 


.287301 
.287977 


CoUng. 


D.  1".      Cotang       M. 


12.30 
12.28 
12.26 
12.24 
12.22 
12.20 
12.18 
12.17 
12.15 
12.13 

12.11 
12.09 
12.07 
12.05 
12.03 
12.01 
12.00 
11.98 
11.96 
11.94 

11.92 
11.90 
11.89 
11.87 
11.85 
11.83 
11.81 
11.79 
11.78 
11.76 

11.74 
11.72 
11.70 
11.69 
11.67 
11.65 
11.64 
11.62 
11.60 
11.58 

11.57 
11.55 
11.53 
11.51 
11.50 
11.48 
11.46 
11.45 
11.43 
11.41 

11.40 
11.38 
11.36 
11.35 
11.33 
11.31 
11.30 
11.28 
11.26 
11.25 

D.  1". 


0.753681 
.752943 
.752206 
.751470 
.750736 
.750002 
.749270 
.748539 
.747809 
.747080 

0.746352 
.745626 
.744900 
.744176 
.743453 
.742731 
.742010 
.741290 
.740571 
.739854 

0.739137 
.738422 
.737708 
.736995 
.736283 
.735572 
.734862 
.734153 
.733445 
.732739 

0.732033 
.731329 
730625 
.729923 
.729221 
.728521 
.727822 
.727124 
.726427 
.725731 

0.725036 
.724342 
.723649 
.722957 
.722266 
.721576 
.720887 
.720199 
.719512 
.718826 

0.718142 
.717458 
.716775 
.716093 
.715412 
.714732 
.714053 
.713376 
.712699 
.712023 
.711348 

Tang.   M. 


•  COSINES,  TANGENTS,  AND  COTANGENTS.        241 

HO                                                              1090 

M. 

Sine 

D.I". 

Cosine. 

D,  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 
1 
2 
3 
4 
5 

9.280599 

.281248 
.281897 
.282544 
.283190 
.283836 

10.82 

10.81 
10.79 
10.77 
10.76 

9.991947 
.991922 
.991897 
-991873 
99184S 
991823 

.41 

.41 
.41 
.41 
.41 

9.288652 
.289326 
.289999 
.290671 
.291342 
.292013 

11.23 
11.22 
11.20 
11.18 
11.17 

0.711348 
.710674 
.710001 
.709329 
.7086C9 
.707987 

60 
59 
58 
57  1 
66 
65 

6 

7 
8 
9 

.284480 
.285124 
.285766 
.286408 

10.74 
10.72 
10.71 
10.69 
10.67 

.991799 
.991774 
.991749 
.991724 

.41 

.41 
.41 

42 

.292682 
.293350 
.294017 
.294684 

11.14 
11.12 
11.11 
11.09 

.707318 
.706650 
.705983 
.705316 

54 
63 
62 
61 

10 
11 
12 
13 

14 

9.287048 
.237688 
.288326 
.288964 
289GOO 

10.66 
10.64 
10.63 
10.61 

9.991699 
.991674 
.991649 
.991624 
.991599 

.42 

.42 
.42 
.42 

9.295349 
.296013 
.296677 
.297339 
.298001 

11.07 

11.H6 
11.04 
11.03 

0.704651 
.703987 
.703323 
.702661 
.701999 

60 
49 
48 
47 
46 

15 
16 
17 
18 
19 

.290236 
.290870 
291504 
.292137 
.292768 

10.59 
10.58 
10.56 
10.55 
10.53 
10.51 

.991574 
.991549 
.991524 
.991498 
.991473 

.42 
.42 
.42 
.42 
.42 

.298662 
.299322 
.299980 
.300638 
.301295 

11.00 
10.98 
10.97 
10.95 
10.93 

.701338 
.700678 
.700020 
.699362 
.698705 

46 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.293399 
.294029 
294658 
295286 
295913 
296539 
..297164 
.297788 
298412 
.299034 

10.50 
10.48 
10.47 
10.45 
10.43 
10.42 
10.40 
10.39 
10.37 
10.36 

9.991443 
.991422 
.991397 
.991372 
.991346 
.991321 
.991295 
.991270 
.991244 
.991218 

.42 
.42 
.42 
.42 
.42 
.43 
.43 
.43 
.43 
.43 

9.301951 
.302607 
.303261 
.303914 
.304567 
.305218 
.305869 
.306519 
.307168 
.307816 

10.92 
16.90 
10.89 
10.87 
10.86 
10.84 
10.83 
10.81 
10.80 
10.78 

0,698049 
697393 
.696739 
.696086 
.695433 
.694782 
.694131 
.693481 
.692832 
.692184 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

S  299655 
300276 
300S95 
.301514 
302132 
.302748 
.303364 
.303979 
.304593 
.305207 

10.34 
10.33 
10.31 
10.30 
10.28 
10.26 
10.25 
10.23 
10.22 
10.20 

9.991193 
.991167 
.991141 
.991115 
.991090 
.991064 
.991038 
.991012 
.990986 
.990960 

.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 

9.308463 
.309109 
.309754 
.310399 
.311042 
.311685 
.312327 
.312968 
.313608 
.314247 

10.77 
10.76 
10.74 
10.73 
10.71 
10.70 
10.68 
10.67 
10.66 
10.64 

0.691537 
.690891 
.690246 
.689601 
.688958 
.688315 
.687673 
.687032 
.686392 
.685753 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
'47 
48 
49 

9.305819 
.306430 
.307041 
.307650 
.308259 
.308867 
.309471 
.310080 
.310685 
.311239 

10.19 
10.17 
10.16 
10.14 
10.13 
10.12 
10.10 
10.09 
10.07 
10.06 

9.990934 
.990908 
.990882 
.990355 
.990829 
.990803 
.990777 
.990750 
.990724 
.990697 

.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 

9.314885 
.315523 
.316159 
.316795 
.317430 
.318064 
.318697 
.319330 
.319961 
.320592 

10.62 
10.61 
10.60 
10.58 
10.57 
10.55 
10.54 
10.53 
10.51 
10.50 

0.685115 

.684477 
683841 
.683205 
.682570 
.681936 
.681303 
.680670 
.680039 
.679408 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 
61 
62 
53 
64 
65 
66 
67 

9.311893 
.312495 
.313097 
313693 
.314297 
.314897 
.315495 
.316092 

10.04 
10.03 
10.01 
10.00 
9.98 
9.97 
9.96 

9.990671 
.990645 
.990618 
.990591 
.990565 
.990538 
.990511 
.990485 

.44 
.44 
.44 
.44 
.44 
.44 
.45 

9.321222 
.321851 
.322479 
.323106 
.323733 
.324358 
324983 
.325607 

10.48 
10.47 
10.46 
10.44 
10.43 
10.41 
10.40 

0.678778 
.678149 
.677521 
.676894 
.676267 
.675642 
675017 
.674393 

10 
9 

8 

7 

68 
69 
60 

316689 
.317284 
.317879 

9.94 
9.93 
9.91 

.99(1458 
.990431 
.990404 

.45 
.45 
.45 

32H231 
326853 
.327475 

10.39 
10.37 
10.36 

.673769 
.673147 
.672525 

M. 

Ocdne. 

D.l». 

8b». 

D.l«. 

Cotang. 

D.  1". 

Tang. 

M. 

1010 


242 


TABLE    II.       LOGARITHMIC    SINES, 


M. 

Sine 

D.I" 

Cosln*. 

D.l". 

Tang. 

D.  1". 

Cotaug. 

M. 

0 

1 

9.317879 
.318473 

9.90 

9.990404 
.990378 

.45 

9.327475 
.328095 

10.35 
in  OQ 

0.672525 
671905 

~60~ 
59 

2 
3 
4 

.319066 
'.320249 

9.87 
9.86 

.990351 
.990324 
.990297 

A5 
.45 

328715 
.329334 
.329953 

lU.oo 

10.32 
10.31 

.671285 
.670666 
.670047 

58 
67 
66 

6 

6 

7 
8 

.320840 
.321430 
.322019 
.322607 

9.S3 
9.81 

9.80 

n  TO 

.990270 

.990243 
.990215 
.990188 

!45 
.45 
.45 

.330570 
.331187 
.331803 
.332418 

m28 
10.27 
10.25 

.669430 
.668813 
.668197 
.667582 

55 

63 
52 

9 

.323194 

y.  /y 

9.77 

.990161 

'.45 

.333033 

10!23 

.666967 

51 

10 
11 

12 
13 
14 

9.323780 
.324366 
.324950 
.325534 
326117 

9.76 
9.75 
9.73 
9.72 

Q  7fl 

9.990134 
.990107 
.990079 
.990052 
990025 

.45 
.45 
.46 
.46 

9.333646 
.334259 
.334871 
.335482 

.33(5093 

1021 

10.20 
10.19 
10.17 
in  IA 

0.666354 
.665741 
.665129 
.664518 
.663907 

60 
49 

48 
47 
46 

16 

.326700 

y.  /  u 

O  £G 

.989997 

.46 

.336702 

I''.  ID 
in  i  K. 

.663298 

45 

16 
17 

.327281 
.327862 

y.oy 
9.68 

Q  A£ 

989970 
989942 

.46 
.46 

AH 

.33731  1 
.337919 

lU.lo 
10.14 

.662689 
.662081 

44 
43 

18 
19 

.328442 
.329021 

y.  DO 
9.65 
9.64 

.989915 

.989887 

.46 
.46 
.46 

.33S527 
.339133 

lo'.n 

10.10 

.661473 
.660867 

42 
41 

20 
21 
22 
23 

9.329599 
.330176 
.330753 
.331329 

9.62 
9.61 
9.60 

9.989860 
.989832 
.989804 
.989777 

.46 
.46 
.46 

9.339739 
.340344 
.340948 
.341552 

10.08 
10.07 
10.06 
in  nK 

0.660261 
.659656 
.659052 
.658448 

40 
39 
38 
37 

24 
25 
26 

27 
28 

.33  '903 
.332478 
.333051 
.333624 
.334195 

9^57 
9.56 
9.54 
9.53 

.989749 
.989721 
.989693 
.989665 
.989637 

.46 
.46 
.46 
.46 
.47 

At 

.342155 
.342757 
343358 
343958 
.344558 

lu.Uo 
10.03 
10.02 
10.01 
10.00 

Q  QQ 

.657845 
.657243 
.656642 
.656042 
.655442 

36 
36 
34 
33 
32 

29 

.334767 

9^50 

.989610 

.47 
.47 

.345167 

y.yo 
9.97 

.654843 

31 

30 

9.335337 

Q  AQ 

9.989582 

9.345755 

Q  QA 

0.654245 

30 

31 
32 
33 
34 
35 
30 

.33591  « 
.336475 
.337043 
.337610 
.338176 
.338742 

y.^y 
9.48 
9.46 
9.45 
9.44 
9.43 

Q  .1  1 

989553 
939525 
989497 
989469 
989441 
989413 

.47 
.47 
.47 

.47 
.47 
.47 

.346353 
.346949 
.347545 
.348141 
.348735 
.349329 

y.yo 
9.95 
9.93 
9.92 
9.91 
9.90 

O  fifi 

.653647 
.653051 
.652455 
.651859 
.651265 
.650671 

29 
28 
27 
26 
25 
24 

37 

38 
39 

.339307 
339871 
.340434 

y.ii 
9.40 
9.39 
9.37 

989385 
989356 
989328 

.47 
.47 
.47 

.47 

.349922 
.350514 
.351106 

y.oo 
9.87 
9.86 
9.85 

.650078 
.649486 
.648894 

23 
22 
21 

4C 
41 

9.340996 
.341558 

9.36 

9989300 
.969271 

.47 

9.351697 
.352237 

9.84 

O  DO 

0.648303 
.647713 

20 
19 

42 

43 
44 
45 
46 

47 
48 
49 

.342119 
.342679 
.343239 
.343797 
.344355 
.344912 
.345469 
.346024 

9.35 
9.34 
9.32 
9.31 
9.30 
9.29 
9.27 
9.26 
9.25 

.989243 
989214 
989186 
989157 
989128 
989100 
.989071 
.989042 

.47 
.47 

.48 
.48 
.48 
.48 
.48 
.48 
.48 

.352876 
.353465 
.354053 
.354640 
.355227 
.355813 
.356398 
.356982 

•LIB 

9.81 
9.80 
9.79 
9.78 
9.76 
9.76 
9.74 
9.73 

.647124 
646535 
.645947 
.645360 
.644773 
.644187 
643602 
.643018 

18 
17 
16 
15 
14 
13 
12 
11 

60 
61 
62 
53 
54 
55 
66 

9.346579 
.347134 
.347687 
.348240 
.348792 
349343 
.349893 

9.24 
9.22 
9.21 
9.20 
9.19 
9.17 

9.989014 

.988985 
.988956 
.988927 
.988898 
.988869 
.988840 

.48 
.48 
.48 
.48 
.48 
.48 

9.357566 
.358149 
.358731 
.359313 
.359893 
.360474 
.361053 

9.72 
9.70 
9.69 
9.68 
9.67 
9.66 

0.642434 
.641851 
.641269 
.640687 
.640107 
.639526 
.638947 

10 
9 

8 
7 
6 
6 
4 

67 

68 

.350443 
.350992 

9.16 
9.15 

91  A 

.988811 
.988782 

.48 
.48 

.361632 
362210 

9.66 
9.63 

.638368 
.637790 

3 

2 

59 

6C 

.351540 
.352088 

.14 
9.13 

988753 
.988724 

.49 
.49 

362787 
363364 

9^61 

.637213 
.636636 

1 

0 

M. 

Coelue. 

D.  1". 

Sine. 

D.  1". 

Cotaug. 

D.l». 

~~Ttag~  "«T 

103° 


770 


COSINES,    TANGENTS,    AND   COTANGENTS. 


243 


M. 

Sine. 

D.  1". 

Cosine. 

D.1". 

Tang. 

D.  1". 

Cotang. 

M. 

Q 

9.352038 

Q  1  1 

9.938724 

49 

9.363364 

o  An 

0.636636 

60 

2 
3 
4 

5 
6 

r 

8 
9 

.352635 
.353131 
.3537?fi 
.354271 
.354315 
.355353 
.355901 
.356443 
.356934 

9*11 

9.10 
9.09 
9.03 
9.07 
9.05 
9.04 
9.03 
9.02 
9.01 

.938695 
.933666 
.983636 
.938607 
.938578 
.988548 
.938519 
.938489 
.938460 

149 
.49 
.49 
.49 
.49 
.49 
.49 

.363940 
.364515 
.365090 
.365664 
.366237 
.366810 
.367382 
.367953 
.363524 

y.ou 
9.59 
9.58 
9.57 
9.55 
9.54 
9.53 
9.52 
9.51 
9.60 

.636060 
.635485 
.634910 
.634336 
.633763 
.633190 
.632618 
.632047 
.631476 

59 
58 
67 
66 
55 
54 
53 
52 
51 

10 

9.357524 

9.938430 

9.369094 

0.630906 

60 

11 
12 
13 
14 
15 
16 
17 
18 
19 

.358064 
.358603 
359141 
.359678 
.360215 
.360752 
.361287 
.361822 
.362356 

a  98 

8.97 
8.96 
8.95 
894 
8.92 
8.91 
8.90 
8.89 

.938401 
.983371 
.983342 
.988312 
.988282 
.988252 
.938223 
.988193 
.983163 

!49 
.49 
.50 
.50 
.50 
.60 
.50 
.60 
.60 

.369663 
•370232 
.370799 
.371367 
.371933 
.372499 
.373064 
.373629 
.374193 

&48 
9.47 
9.45 
9.44 
9.43 
9.42 
9.41 
9.40 
9.39 

.630337 
.629768 
.629201 
.628633 
.628067 
.627501 
.626936 
.626371 
.625807 

49 
48 
47 
46 
45 
44 
43 
42 
41 

20 

9.362889 

Q  OQ 

9.988133 

en 

9.374756 

O  OQ 

0.625244 

40 

21 
22 
23 
24 
26 
'26 
27 
28 
29 

.363422 
.363954 
.364485 
.365016 
.365546 
.366075 
.3G6604 
.367131 
.367659 

O.oo 

8.87 
8.36 
8.84 
8.83 
8.82 
8.81 
8.80 
8.79 
8.78 

.988103 

.988073 
.988043 
.988013 
.937983 
.S37953 
.987922 
987892 
987862 

.OiJ 

.50 
.60 
.50 
.50 
.60 
.50 
.50 
.60 
.61 

.375319 
.375831 
.376442 
.377003 
.377563 
.378122 
.378681 
.379239 
.379797 

y.  oo 
9.37 
9.36 
9.35 
9.33 
9.32 
9.31 
9.30 
9.29 
9.28 

.624681 
.624119 
.623558 
.622997 
.622437 
621878 
.621319 
.620761 
.620203 

39 
38 
37 
36 
35 
34 
33 
32 
31 

30 

9.363185 

o  70 

9.987832 

9.380354 

0.619646 

30 

31 

368711 

o.  /o 

Q  -YE 

987801 

C| 

.380910 

O  OA 

.619090 

29 

32 
33 

369236 
369761 

O.  /D 

8.74 

Q  P*O 

987771 
987740 

.01 

.61 

Cl 

.381466 

.332020 

y.to 
9.25 

.618534 
.6179.30 

28 
27 

34 
35 
36 
37 

370285 
370308 
371330 
371852 

O.ld 

8.72 
8.71 
8.70 

.987710 
.987679 
987649 
.987618 

.51 
.51 
.51 
.61 

C| 

.382575 
.383129 
.383632 
.384234 

9/23 
9.22 
9.21 
Q  on 

.617425 
.616871 
.616318 
.615766 

26 
25 
24 
23 

38 
39 

.372373 
.372394 

8'63 
8.66 

.987588 
.987557 

.91 

.61 
.61 

.384786 
.385337 

«7.<CU 

9.19 
9.18 

.615214 
.614C63 

22 
21 

40 
41 
42 
43 
44 
1  45 
i  46 
47 
43 

9.373414 
.373933 
.374452 
.374970 
.375487 
.376003 
.376519 
.377035 
.377549 
.378063 

8.65 
8.64 
8.63 
8.62 
8.61 
8.60 
8.59 
8.58 
8.57 
8.56 

9.987526 
.987496 
.937465 
.987434 
.987403 
.987372 
.987341 
.937310 
.987279 
.987248 

.61 
.51 
,61 
.51 
.51 
.52 
.52 
.52 
.62 
.62 

9.385888 
.386438 
.386987 
.387536 
.383084 
.388631 
.389178 
.389724 
.390270 
.390815 

9.17 
9.16 
9.16 
9.14 
9.12 
9.11 
9.10 
9.09 
9.08 
9.07 

0.614112 
.613562 
.613013 
.612464 
.611916 
.611369 
.610822 
.610276 
.609730 
.609185 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60* 
51 
52 

9378577 
.379089 
.379601 

8.55 
8.53 

Q  CO 

9.987217 
.987186 
.987155 

52 

.52 

9.391360 
.391903 
.392447 

9.06 
9.05 

0.608640 
.608097 
.607553 

10 
9 

8 

53 

.380113 

(MBI 

.987124 

.52 

.392989 

9.04 
Q  nQ 

.607011 

54 
55 
56 
57 

58 

.380624 
;331134 
.381643 
.332152 
.382661 

a  so 

8.49 

8.48 
8.47 

.987092 
.987061 
.987030 
.986998 
.986967 

^52 
.52 
.52 
52 

.393531 
.394073 
.394614 
.395154 
.395694 

tf.Uo 
9.02 
9.01 
9.00 
8.99 

.606469 
.605927 
.605386 
.604346 
.604306 

59 
60 

.333163 
.333675 

a  45 

.986936 
.936904 

'.52 

.396233 
.396771 

8'97 

.603767 
.603229 

1 
0 

|M. 

Cosine. 

D.I'. 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

M. 

244            TABLE  II.   LOGARITHMIC  SINES, 
1*°                                            1660 

M 

Bine. 

D.  1". 

CoBlne. 

D.l". 

Tang. 

D.  1". 

Cotang 

M. 

b 

1 
2 

9.383675 
.384182 

.3846-87 

8.44 
8.43 

9.986914 
.986873 
.986.841 

.53 
.53 

9.396771 
.397309 
.397846 

8.96 
8.96 

Q  QC 

0.603229 
.602691 
.602154 

~sb~ 

59 

58 

3 

.385192 

8.4< 

.986809 

.o3 

.393383 

O.95 

.601617 

57 

4 

5 
6 

7 

.385697 
.336201 
.336704 
.387207 

8.41 

8.40 
8.39 
8.33 

.986778 
.936746 
.936714 
.936683 

.53 
.53 
.53 
.53 

.393919 
.399455 
399990 
.40D524 

8.94 
8.93 
8.92 
8.91 

.601031 
.600545 

.600(110 
.599476 

56 
55 
54 
53 

8 

.387709 

8.37 

.9S66.-»1 

.53 

.401058 

8.90 

.593942 

52 

9 

.388210 

8.36 
8.35 

.986619 

.53 
53 

.401591 

8.89 
8.88 

.598409 

51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.388711 
.33921  1 
.38971  1 
.390210 
.390703 
.391206 
.391703 
.392193 
.392695 
.393191 

8.34 

8.33 
8.32 
8.31 
8.30 
8.29 
8.28 
8.27 
8.26 
8.25 

9.9S6537 
.986555 
.936523 
.936491 
.986459 
.9,36427 
.936395 
.9S6363 
.936331 
.986299 

.53 
.53 
.53 
.53 
.53 
.64 
.54 
.64 
.54 
.54 

9.402124 
.402656 
.403187 
.403718 
.404249 
.404778 
.405308 
.405836 
.406364 
.406392 

8.87 
8.86 
8.85 
8.84 
8.83 
8.82 
8.81 
8.80 
8.79 
8.78 

0.597876 
.597344 
.B96813 
.596282 
.595751 
.595222 
.594692 
.594164 
.593636 
.693108 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 

9.393685 
.394179 
.394673 
.395  1  6G 
.395653 
.396150 

8.24 

8.23 
8.22 
821 
8.20 

9.9S6266 
.9S6234 
.9,36202 
.936169 
.936137 
.936104 

.54 
.54 
.64 
.54 
.54 

9.407419 
.407945 
.408471 
.408996 
.409521 
.410045 

8.77 
8.76 
8.75 
8.75 
8.74 

0.592581 
.592055 
.591529 
.591004 
.690479 
.589955 

40 
39 
38 
37 
36 
35 

26 

.396641 

8.19 

.936072 

.54 

.410569 

8.73 

Q  TO 

.689431 

34 

27 

.397132 

8.  18 

.986039 

.54 

.411092 

a.  74 

'.588908 

33 

28 

.397621 

8.17 

.986007 

.54 

.411615 

8.71 

.588385 

32 

29 

.398111 

8.18 
8.15 

.985974 

.54 
.54 

.412137 

8.70 
8.69 

.687863 

31 

30 

9.398600 

9.985942 

9.412658 

0.687342 

30 

31 
32 
33 

.399038 
.399575 

.400062 

8.14 
8.13 
8.12 

.935909 
.985876 
.985843 

.64 

.55 
.55 

.413179 
.413699 
.414219 

8.68 
867 
8.66 

.586821 
.586301 

.585781 

29 

28 
27 

34 
35 
36 
37 
33 
39 

.400549 
.401035 
.401520 
402005 
.402439 
.402972 

8.11 
8.10 
8.09 
8.03 
8.07 
8.06 
8.05 

.985811 
.985778 
.985745 
.985712 
.985679 
.985646 

.55 
.55 
.55 
.65 
.55 
.55 
.55 

.4H738 
.415257 
.415775 
.416293 
.416310 
.417326 

8.65 
8.65 
8.64 
8.63 
8.62 
8.61 
8.60 

.585262 
.584743 
.584225 
.583707 
.583190 
.582674 

26 
25 
24 
23 
22 
21 

40 

9.403455 

9.985613 

9.417842 

0.582158 

20 

41 

.403933 

8.04 

.935530 

.55 

.418358 

8.59 

o  ro 

.581642 

19 

42 

.404420 

8.03 

.985547 

.55 

.418873 

o.5o 

.581127 

18 

43 
44 
45 

46 
47 

48 

.404901 
.405332 
.405862 
.406341 
.406320 
.407299 

8.02 
8.01 
800 
799 
7.98 
7.97 

.935514 
.935430 
.985447 
.985414 
.935331 
.985347 

.55 
.55 
.55 
.55 
.56 
.56 

.419387 
.419901 
.420415 

.420927 
421440 
421952 

8.57 
8.56 

a  56 

8.55 
8.54 
8.53 

.580613 
.580099 
.579585 
.679073 
.578560 
.678043 

17 
16 
15 
14 
13 
12 

49 

.407777 

7.96 
7.96 

.935314 

.56 
.56 

.422463 

8.52 
8.51 

.577537 

11 

50 
51 

9.408254 
.408731 

7.95 

*»  a  * 

9.985230 
.985247 

.56 

9.422974 
.423484 

8.50 

0.577026 
.576516 

10 
9 

52 

.409207 

.  .94 

.985213 

.56 

.423993 

8.49 

.576007 

8 

53 
54 
55 

.409632 
.410157 
.410632 

7.93 
7.92 
7.91 

.935180 
.935146 
.985113 

.56 
.56 
.56 

.424503 
.42501  1 
.425519 

8.49 

8.43 
8.47 

.575497 
.574989 
.574431 

7 
6 
5 

56 

.411106 

7.90 

.985079 

.56 

.426027 

8.46 

.573973 

4 

67 

53 
59 

.411579 
.412052 
.412524 

7.89 
7.88 
7.87 

.985045 
.98501  1 
.984978 

.56 
.56 
.56 

426534 
427(141 
.427547 

8.45 
8.44 
8.43 

.573466 
.572959 
.572453 

3 

a 

i 

60 

.412996 

7.86 

.984944 

.56 

.423052 

8.43 

.571948 

0 

M. 

Coelue. 

D.l» 

Sine. 

D.  1". 

Cotang. 

D.1". 

Tang. 

M. 

104Q 


75 


COSINES,    TANGENTS,    AND   COTANGENTS. 


245 


150 


M. 

Sine. 

D.l". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 
1 
2 
3 

9.412996 
.413467 
.413938 
414408 

7.85 
7.84 
7.84 

9.984944 
.984910 
.984876 
984842 

.56 
.57 
.57 

9.428052 
428558 
429062 
.429566 

8.42 
8.41 
8.40 

0.571948 
.571442 
.570938 
.670434 

60 
59 

58 
57 

4 

6 

414873 
.415347 

7.82 

984808 
984774 

57 

430070 
430573 

8.38 

Q  OQ 

.569930 
.669427 

56 
55 

6 

7 

.415815 
.416283 

7.80 

984740 

984706 

57 

.431075 
431577 

8.37 

.568925 
.668423 

54 
63 

8 
9 

.416751 
.417217 

7.78 
7.77 

984672 
984638 

57 
57 

.432079 
.432580 

8.35 
8.34 

.567921 
.667420 

52 
51 

10 
11 
12 
13 
14 
16 
16 

9.417684 
.418150 
.418615 
.419079 
.419544 
.420007 
420470 

7.76 
7.75 
7.75 
7.74 
7.73 
7.72 

9.984603 
984569 
984535 
.984500 
.984466 
.984432 
.984397 

.67 
.67 
.67 
.67 
.67 
.67 

9.433080 
433580 
.434080 
434579 
.435078 
.435576 
.436073 

8.33 
8.33 
8.32 
8.31 
8.30 
8.29 

0.566920 
.566420 
.665920 
.665421 
.564922 
.564424 
.563927 

50 
49 
48 
47 
46 
45 
44 

17 
18 
19 

420933 
.421395 

.421857 

7.70 
7.69 
7.68 

.984363 
.984328 
.934294 

.68 
.68 
.68 

.436570 
.437067 
.437563 

8.28 
8.27 
8.26 

.563430 
.562933 
.662437 

43 
42 
41 

20 
21 

9.422318 

422778 

7.67 

9.984259 

.984224 

.68 

9.438059 
438554 

8.25 

0.561941 
.561446 

40 
39 

22 
23 
24 

25 
26 
27 

28 

123238 
423697 
424156 
424615 
425073 
425530 
425987 

7.67 
7.66 
7.65 
7.64 
7.63 
7.62 
7.61 

.984190 
.984155 
.984120 
.984085 
.984050 
934015 
983981 

.68 
.68 
.68 
.68 

.68 
.68 

439048 
.439543 
.440036 
440529 
441022 
441514 
442006 

8.24 
8.23 
8.22 
8.21 
8.20 
8.20 

.560952 
560457 
.659964 
.659471 
.558978 
658486 
.557994 

38 
37 
36 
36 
34 
33 
32 

29 

426443 

7.61 
760 

.983946 

.68 
68 

442497 

8.18 

.657503 

31 

30 

9.426899 

9.9839  H 

9.442988 

0.557012 

30 

31 
32 
33 
34 
35 
36 
37 
38 
59 

427354 
427809 
428263 
428717 
429170 
429623 
430075 
430527 
430978 

7.58 
7.57 
7.56 
7.55 
7.55 
753 
7.52 
7.52 
751 

.983875 
.983840 
.983805 
983770 
983735 
983700 
983664 
983629 
983594 

.68 
.68 
69 
.69 
.69 
.59 
.69 
.69 
.59 
69 

443479 
443968 
444458 
444947 
445435 
445923 
446411 
.446898 
447384 

8.16 
8.16 
8.15 
8.14 
8.13 
8.13 
8.12 
8.11 
8.10 

.556521 
.556032 
.555542 
.555053 
.554565 
.554077 
.553589 
.553102 
.652616 

29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 

9.431429 
431879 
432329 

7.60 
7.49 

9.983558 
983523 
983487 

59 
.69 

9.447870 
448356 
448841 

8.09 
8.09 

0.552130 
.551644 
.651159 

20 
19 

18 

13 

432778 

983452 

.69 

449326 

.550674 

M 

45 
16 
47 

48 
49 

433226 
433G75 
434122 
434569 
435016 
435462 

7.47 
746 
7.45 
7.44 
7.44 
7.43 

983416 
933381 
983345 
983309 
983273 
983233 

.69 
.59 
.59 
.69 
.60 
.60 
.60 

449810 
450294 
450777 
451260 
451743 
452225 

8.06 
8.06 
8.05 
8.04 
8.03 
8.03 

.550190 
.549706 
.549223 
.548740 
.548257 
.547775 

16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 
55 
56 
57 
68 

9435908 
436353 
436793 
437242 
437686 
438129 
438572 
439014 
.439456 

7.42 
7.41 
7.40 
7.40 
7.39 
7.38 
7.37 
736 

9.9832(12 
983166 
983130 
983094 
.983058 
.983022 
982986 
982950 
.982914 

.60 
.60 
.60 
.60 
.60 
.60 
.60 
.60 

9.452706 

453187 
.453663 
454148 
454628 
455107 
455586 
456064 
456542 

8.02 
8.01 
8.00 
8.00 
7.99 
7.98 
7.97 
7.97 

0.  f  47294 
.546813 
.546332 
.545852 
.545372 
.544893 
.544414 
.543936 
.543458 

10 
9 
8 
7 
6 
6 
4 
3 
2 

59 
60 

.439397 
|  .440338 

7  36 
7.35 

.982878 
.982842- 

.60 
60 

457019 
457496 

7.95 

.542981 
.542504 

1 
0 

M.  |  Oceinu. 

D.l". 

81ne. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

M. 

246            TABLE  II.   LOGARITHMIC  SINES, 
100                                            1030 

M. 

Sine. 

D  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

9.440338 

9.982842 

9.457496 

0.542504 

60 

1 
2 
3 
4 
5 
6 
7 

.440778 
.441218 
.441658 
.442096 
.442535 
.442973 
.443410 

7.34 
7.33 
7.32 
7.31 
7.31 
7.30 
7.29 

.982805 
.982769 
.93*733 
.982696 
.982660 
.982624 
.982587 

'.60 
.61 
.61 

61 

.61 
.61 

fit 

.457973 

.45,3449 
.458925 
.459400 
.459375 
.460349 
.460323 

7^94 
7.93 
7.92 
7.91 
7.91 
7.90 

.542027 
.541551 
.541075 
.540600 
.540125 
.539651 
.539177 

59 
68 
57 
56 
55 
54 
53 

8 
9 

.443347 
.444234 

7.27 
7.87 

.982551 

.982514 

.01 
.61 

.61 

.461297 
.461770 

7.88 
7.88 

.638703 

.538230 

52 
51 

10 
11 
12 
13 
14 
15 
16 
17 

9.444720 
.445155 
.445590 
.446025 
.446459 
.446893 
.447326 
.447759 

7.26 
7.25 
7.24 
7.24 
7.23 
7.22 
7.21 

9.932477 
.982441 
.982404 
.932367 
.982331 
.982294 
.982257 
.982220 

.61 
.61 
.61 
.61 
.61 
.61 
.61 

9.462242 
.462715 
.463186 
.463658 
.464128 
.464599 
.465069 
465539 

7.87 
7.86 
7.86 
7.85 
7.84 
7.83 
7.83 

ff  QO 

0.637758 
.537285 
.636814 
.536342 
.535872 
.535401 
.534931 
.534461 

50 
49 

48 
47 
46 
45 
44 
43 

18 
19 

.448191 
.448623 

7.20 
7.20 
7.19 

.982183 
.932146 

!ea 

.62 

.466008 
.466477 

f  *9Si 

7.81 
7.81 

.533992 
.533523 

42 
41 

20 
21 
22 

9.449054 
.449435 
.449915 

7.18 
7.17 

9.932109 
.982072 

.982035 

.62 

.62 

9.466945 
.467413 
.467880 

7.80 
7.79 

7  T'Q 

0.533055 
.532587 
632120 

40 
39 
38 

23 

450345 

7.17 

71  ii 

.981998 

'S 

.468347 

7.  /  O 
7  7fl 

.631653 

37 

24 

.450775 

.  10 

.981961 

RO 

.468814 

/./C5 

.631136 

36 

25 
26 
27 
28 
29 

.451201 
.451632 
.452060 
.452438 
.452915 

7.15 
7.14 
7.13 
7.13 
7.12 
7.11 

.931924 
.9818^6 
.931849 
.931^12 
.981774 

'.62 
.62 
.62 
.62 
.62 

.469280 
.469746 
.47021  1 
.470676 
.471141 

7>6 
7.76 
7.75 
7.74 
7.74 

.530720 
.530254 
.529789 
.529324 
.528859 

35 
34 
33 
32 
31 

30 

9.453342 

9.981737 

9.471605 

7  7O. 

0.528395 

30 

31 
32 
33 
34 
35 

.453768 
.454194 
.454619 

.455044 
.455469 

7.10 
7.10 
709 
7.08 
7.07 

.931700 
.981662 
.981625 
.981587 
.931549 

!62 
.63 
.63 
.63 

CQ 

.472069 
.472532 
.472995 
.473457 
.473919 

/.  to 
7.72 
7.71 
7.71 
7.70 

.627931 
.527468 
.527005 
.526543 
.526081 

29 
28 
27 
26 
25 

36 
37 
38 
39 

.455893 
.456316 
.456739 
.457162 

7.07 
7.06 
7.05 
7.04 
7.04 

.981512 
.931474 
.931436 
.931399 

.DO 

.63 
.63 
.63 
.63 

.474381 
.474842 
.475303 
.475763 

7.69 
7.63 
7.67 
7.67 

.525619 
.625158 
.524697 
.624237 

24 
23 
22 
21 

40 
41 

9.457584 

.458006 

7.03 

9.931361 
.981323 

.63 

9.476223 

.476633 

7.66 

0.523777 
.523317 

20 
19 

42 

.458427 

7.02 

.981235 

.63 

.477142 

7.65 

.522358 

18 

43 
44 

.458348 
459268 

7.01 
701 

.981247 

.981209 

'.63 

/»Q 

.477601 
.478059 

7.65 
7.64 

T  £O 

.522399 
.521941 

17 
16 

45 
46 

.459688 
.460108 

7.00 
6.99 

.981171 
.981133 

.DO 

.63 

CO 

.478517 

.478975 

7.6,3 
7.63 

.521483 
.521025 

16 
14 

47 
48 
49 

.460527 
.460946 
.461364 

6.98 
6.98 
6.97 
6.96 

.981095 
.981057 
.981019 

.OO 

.64 
.64 
.64 

.479432 

.479889 
.480345 

7^61 
7.61 
7.60 

.520568 
.520111 
.519655 

13 
12 
11 

50 
51 

9.461782 
.462199 

6.96, 

9.980981 
.9,30942 

.64 

CA 

9.480801 
.481257 

7.59 

ff  c<\ 

0.519199 
.618743 

10 
9 

52 

.462616 

6.95 

.980904 

.64 

.481712 

7.59 

.518288 

8 

53 
64 
55 
56 
57 
58 
59 
60 

.463032 
.463448 
.463364 
.464279 
.464694 
.465108 
.465522 
.465935 

6.94 
6.93 
6.93 
692 
6.91 
6.90 
690 
6.89 

.980366 
.980827 
.980789 
.980750 
.980712 
.980673 
.980635 
.980596 

.64 
.64 
.64 
.64 
.64 
.64 
.64 
.64 

.482167 
.482621 
.483075 

'.483982 
.484435 
.484887 
.485339 

"*.58 
7.57 
7.57 
7.56 
7.55 
755 
7.54 
7.53 

.517833 
.617379 
.516925 
.516471 
.616018 
.515565 
.515113 
.514661 

7 
6 
6 
4 
3 
2 
I 
0 

M. 

Cosine. 

D.  1". 

Sin* 

D.I". 

Cotang. 

D.  1". 

Tang. 

M. 

COSINES,    TANGENTS,    AND   COTANGENTS. 


247 


M. 

Sine. 

D.I". 

Ooalne. 

D.  1". 

Tang. 

D.  I". 

Cotang. 

M. 

0 
1 

9.465935 
466348 

6.83 

A  QQ 

9.980596 

.930558 

.64 

fid 

9.485339 
.485791 

7.53 

ry  CO 

0.514661 
.514209 

60 
59 

2 
3 
4 
6 

466761 
.467173 
467585 
467996 

o.oo 
6.87 
6.86 
6.85 

A  QK. 

.930519 

.980480 
.980442 
.980403 

.l>4 

.65 
.65 
.65 

.486242 
.486693 
.487143 
.487593 

7.54 
7.51 
7.51 
7.50 

.513753 
.513307 
.512357 
.512407 

58 
57 
56 
55 

6 

7 
8 
9 

.468407 
.463817 
.469227 
.469637 

D.ou 

6.84 
6.83 
6.83 
6.82 

.980364 
.930325 
.980236 
.980247 

'.65 
.65 
.65 
.65 

.483043 
.488492 
.488941 
.489390 

7.50 
7.49 
7.43 
7.48 
7.47 

.511957 
.511508 
.511059 
.510610 

54 
53 
52 
51 

10 
11 
12 

9.470046 
.470455 
.470863 

6.81 
6.31 

£  on 

9.980208 
.980169 
.980130 

.65 
.65 

AC 

9.439838 
.490286 
.490733 

7.46 
7.46 

0.510162 
.509714 
.509267 

50 
49 

48 

13 
14 

.471271 
.471679 

D.oU 
679 

A  TO 

.980091 
.930052 

.DO 

.65 

.491180 
.491627 

7.45 
7.44 

.508820 
.508373 

47 
46 

15 
16 
17 

.472086 
.472492 
472898 

O-/O 

6.78 
677 

.9,30012 
.979973 
.979934 

.65 
.65 

.492073 
.492519 
.492965 

7.44 
7.43 
7.43 

.507927 
.507481 
.507035 

45 
44 
43 

18 
19 

473304 
473710 

6.76 
676 
6.75 

.979895 
.979855 

.66 
.66 
.66 

.493410 
.493354 

7.42 
7.41 
7.41 

.506590 
.506146 

42 
41 

20 
21 
22 
23 

9.474115 
474519 
474923 
475327 

6.74 
6.74 
6.73 

9.979316 
.979776 
.979737 
.979697 

.66 
.66 
.66 

9.494299 
.494743 
.495186 
.495630 

7.40 
7.39 
7.39 

0.505701 
.505257 
.504814 
.504370 

40 
39 
33 
37 

24 

25 
26 

475730 
476133 
476536 

6.72 
6.72 
671 

.979658 
.979618 
.979579 

.66 
.66 
.66 

.496073 
.496515 
.496957 

7.38 
7.38 
7.37 

.503927 
.503485 
.503043 

36 
35 
34 

27 

.476938 

6.70 

.979539 

.66 

497399 

7.36 

.502601 

33 

28 

29 

.477340 
.477741 

6.69 
6.69 
6.68 

.979499 
.979459 

.66 
.66 
.66 

.497841 
.498282 

7.36 
7.35 
7.34 

.502159 
.501718 

32 
31 

30 

9.478142 

9.979420 

9.498722 

0.501278 

30 

31 

.478542 

6.67 

.979380 

66 

499163 

7.34 

.500837 

29 

32 
33 
34 
35 
36 
37 
38 

.478942 
.479342 
.479741 
.480140 
.480539 
.430937 
.481334 

6.67 
6.66 
665 
6.65 
6.64 
6.63 
6.63 

.979340 
.979300 
.979263 
.979220 
.979180 
.979140 
.979100 

.66 
.67 
.67 
.67 
.67 
.67 
.67 

.499603 
.500042 
.500481 
.500920 
.501359 
.501797 
.502235 

7.33 
7.33 
7.32 
7.31 
7.31 
7.30 
7.30 

.500397 
.499958 
.499519 
.499080 
.498641 
.498203 
.497765 

28 
27 
26 
25 
24 
23 
22 

39 

.481731 

6.62 
6.61 

.979059 

.67 
.67 

502672 

7.29 
7.28 

.497328 

21 

40 

9.482128 

9.979019 

9.503109 

0.496391 

20 

1  41 

.432525 

6.61 

.978979 

.67 

503546 

7.28 

.496454 

19 

42 
43 

.432921 
.483316 

6.60 
6.59 

.978939 
.973898 

.67 
.67 

503982 
.504418 

7.27 
7.27 

.496018 
.495582 

18 
17 

44 

.483712 

6.59 

.973858 

.67 

.504854 

7.26 

.495146 

16 

45 
46 
47 

.484107 
.484501 
.434395 

6.58 
6.57 
6.57 

.978817 
.978777 
.978737 

.67 
.67 
.67 

.505239 
.505724 
.506159 

7.25 
7.25 
7.24 

.494711 
.494276 
.493341 

15 
14 
13 

48 

.485239 

6.56 

.978696 

.68 

.506593 

7.24 

.493407 

12 

49 

.485682 

6.55 
6.55 

.978655 

.68 
68 

.507027 

7.23 
7.23 

.492973 

11 

50 

9.486075 

9.978615 

9.507460 

0  492540 

10 

51 

.486467 

6.54 

.978574 

.68 

.507893 

7.22 

.492107 

9 

52 

.486860 

6.54 

.973533 

.63 

.503326 

7.21 

.491674 

8 

53 

.437251 

6.53 

.973493 

.68 

.503759 

7.21 

.491241 

7 

54 
55 
56 

.437643 

.438034 
.48,3424 

6.52 
6.52 
6.51 

.973452 
.978411 
.978370 

68 
.68 
.63 

.509191 
.509622 
.510T54 

7.20 
7.20 
7.19 

.490809 
.490378 
.439946 

6 
5 
4 

57 
53 
59 

.488814 
.489204 
.489593 

6  50 
6.50 
6  19 

.978329 

.978288 
.978247 

.68 
.68 
.63 

.510435 
.510916 
.511^46 

MS 
7.17 

.489515 
.489034 

.488654 

3 
2 

1 

60 

.489982 

6.48 

.973206 

.68 

.511776 

7.17 

.4^8224 

0 

M. 

Cosine. 

D.  1". 

Sloe. 

D.  1" 

Cotang. 

D.  1". 

Tang. 

1ST 

73° 


248             TABLE  11.   LOGARITHMIC  SINES, 

180                                            161° 

M. 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Taog. 

D.  1". 

Cotang. 

M. 

f 

9.489982 
.490371 

6.48 

9.978206 
.978165 

.68 

CO 

9.511776 
.5122(16 

7.16 

71  ft 

0.488224 
.487794 

~60~ 
59 

2 

.490759 

ii.47 

.978124 

.oy 

.512635 

•  lo 

71  r 

.487365 

58 

3 

.491147 

6.46 

.978083 

.69 

.513064 

.15 

7  U   • 

486936 

57 

4 

.491535 

6.46 

.978042 

CO 

513493 

i-jj    186507 

56 

6 
6 
7 

8 
9 

.491922 
.492308 
.492695 
.493081 
.493466 

6.4o 
645 
644 
643 
643 
6.42 

.973001 
.977959 
.977918 
.977877 
.977835 

.oy 
.69 
.69 
.69 
.69 
.69 

.513921 
.514349 
.514777 
.615204 
.515631 

7.14  • 

7.13 
7.13 
7.12 
7.12 
7.11 

186079 
.485651 
485223 
.484796 
.484369 

66 
64 
53 
52 
61 

10 

9.493851 

&  A\ 

9.977794 

9.516057 

7  in 

0.483943 

50 

11 

.494236 

0.41 

.977752 

.69 

.616484 

.111 

.483516 

49 

12 
13 
14 
15 
16 
17 
18 
19 

.494621 
.495005 
.495383 
.495772 
.496154 
.496537 
.496.H9 
.497301 

6.41 
6.40 
6.39 
6.39 
6.33 
6.38 
6.37 
636 
6.36 

.977711 
.977669 
.977623 
.977586 
.977544 
.977503 
.977461 
.977419 

.69 
.69 
69 
.69 
.69 
.70 
.70 
.70 
.70 

.61-6910 
517335 
.517761 
.518186 
.518610 
.519034 
.619458 
.519882 

7.  10 
7.09 
7.09 
7.08 
7.08 
7.07 
7.07 
7.06 
7.05 

.483090 
.482665 
.482239 
.481814 
.481390 
.480966 
.480542 
.480118 

48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 

9.497682 

.498064 
.498444 

.498825 

6.35 
6.34 
6.34 

A  oo 

9.977377 
.977335 
.977293 
.977251 

.70 
.70 
.70 

9.520305 
.520723 
.521151 
.621573 

7.05 
7.04 
7.04 

T  fiQ 

0.479695 
.479272 
.478849 
.478427 

40 
39 
38 
37 

24 
25 

.499204 
.499584 

D.OO 

6.33 

£  QO 

.977209 
.977167 

.70 

.521995 
.522417 

/.Uo 

7.03 

T  ftf) 

.478005 
.477583 

36 
36 

26 
27 

28 
29 

.499963 
.500342 
.500721 
.501099 

O.O* 

6.31 
6.31 
6.30 
6.30 

.977125 
.977083 
.977041 
976999 

.70 
.70 
.70 
.70 

.522838 
.523259 
.523680 
.524100 

7.IW 

7.02 
7.01 

r.oi 

7.00 

.477162 
.476741 
.476320 
.475900 

34 
33 
32 
31 

30 
31 

9.501476 
.501854 

629 

9.976957 
.976914 

.70 

9.524520 

.524940 

6.99 

0.476480 
.476080 

30 
29 

32 

.502231 

6.23 

.976872 

.525359 

£  QQ 

.474641 

28 

33 
34 
36 
36 
37 
38 
39 

.502607 
.502984 
.503360 
.503735 
.504110 
.504485 
.504860 

6.23 
6.27 
6.27 
626 
625 
6.25 
6.24 
6.24 

.976830 
.976787 
.976745 
.976702 
.976660 
.976617 
.976574 

.71 
.71 
.71 
.71 
.71 
.71 
.71 
.71 

.625778 
626197 
.526615 
.527033 
.527451 
.627863 
.528285 

6  yy 
6.98 
6.97 
6.97 
6.96 
6.96 
6.C5 
6.95 

.474222 
.473803 
.473385 
472967 
472549 
.472132 
.471716 

27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.505234 
.505608 
.505981 
.506354 
.506727 
.507099 
.507471 
.607843 
.508214 
.508585 

6.23 
622 
622 
6.21 
6.21 
6.20 
6.19 
6.19 
6.18 
6.18 

9.976532 
.976489 
.976446 
.976404 
.976361 
.976318 
.976275 
.976232 
.976189 
.976146 

.71 
.71 
.71 
.71 
.71 
.72 
.72 
.72 
.72 
.72 

9.528702 
.529119 
.529535 
.529951 
.530366 
.630781 
.531196 
.531611 
.532025 
.632439 

6.94 
6.94 
6.93 
6.93 
6.92 
6.91 
6.91 
6.90 
6.90 
6.89 

0.471298 
.470881 
.470465 
.470049 
.469634 
.469219 
.463804 
.468389 
467975 
.467561 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 

9.508956 

9.976103 

9.532853 

£  on 

0.467147 

10 

61 
52 
53 
54 
55 
56 
57 
63 
59 
60 

.509326 
.50%96 
.510065 
.510434 
.510803 
.511172 
.511540 
.511907 
.512275 
.512642 

6.17 
6.16 
6.16 
6.15 
6.15 
6.14 
6.14 
6.13 
6.12 
6.12 

.976060 
.976017 
.975974 
.975930 
.975887 
.975844 
.975800 
.975757 
.975714 
.975670 

.72 

.72 
.72 
.72 
.72 
.72 
.72 
.72 
.72 
.72 

.533266 
.533679 
.534092 
.534504 
.534916 
.535328 
.535739 
.536150 
536561 
.536972 

o.  By 

6.88 
6.88 
6.87 
6.87 
6.86 
6.86 
6.85 
6.85 
6.84 

.466734 
.466321 
465908 
.465496 
.465084 
.464672 
464261 
.463850 
.463439 
.463028 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Coeiuo. 

D.1". 

Bine. 

D.  F. 

Cotang. 

D.  !'•. 

Tang. 

M. 

108° 


COSINES,  TANGENTS,  AND  COTANGENTS.        24f 

190                                              160° 

U 

Sine. 

D.  1". 

Coelne. 

D.l" 

Tang. 

D.  1". 

Cota.ig 

! 

M. 

9.512642 

61  1 

9.975670 

70 

9.536972 

£  QA 

0.46?028 

60 

.513009 
.513375 
.513741 
.514107 

.1  1 
6.11 
6.10 

6.09 

.97.0627 
.975583 
.975569 
.975496 

./O 

.73 
.73 
.73 
73 

.537382 
.537792 
.538202 
.538611 

D.B4 
6.83 
6.S3 
6.82 
a  QO 

.462618 
.462208 
.461798 
.461389 

59 
58 
57 
56 

7 
8 
9 

.614472 
.514837 
.515202 
515566 
615930 

6.08 
6.08 
6.07 
6.07 
6.06 

.975452 
.975408 
.975365 
.975321 
.975277 

'.73 
.73 
.73 
.73 
.73 

.539020 
.539429 
.539837 
.540245 
.540653 

O.O'G 

6.81 
6.81 
6.80 
6.80 
6.79 

.460980 
.460571 
.460163 
.459755 
.459347 

55 
54 
53 
52 
51 

10 
11 
12 
13 
14 
15 
16 
17 
13 
19 

3.516294 
.516657 
.517020 
.517382 
.617745 
.518107 
.518468 
.518829 
.519190 
.519551 

6.05 
6.05 
6.04 
6.04 
6.03 
6.03 
6.02 
6.02 
6.01 
6.00 

9.975233 
.975189 
.975145 
.975101 
.975057 
.975013 
.974969 
.974925 
.974880 
.974836 

.73 
.73 
.73 
.73 
.73 
.74 
.74 
.74 
.74 
.74 

9.541061 
.541468 
.541875 
.642281 
.542688 
.543094 
.543499 
.543905 
.544310 
.644715 

6.79 
6.78 
6.78 
6.77 
6.77 
6.76 
6.76 
6.75 
6.75 
6.74 

0.458939 
.458532 
.458125 
.457719 
.457312 
.456906 
.456501 
.456095 
.455690 
.455285 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

9.519911 
.520271 
.520631 
.520990 
.521349 
.521707 
.522066 
.522424 
.522781 
.523138 

6.00 
5.99 
6.99 
5.98 
5.98 
5.97 
5.97 
5.96 
5.95 
6.95 

9.974792 
.974748 
.974703 
.974659 
.974614 
.974570 
.974525 
.974481 
.974436 
.974391 

.74 
.74 
.74 

.74 
.74 
.74 
.74 
.74 
.74 
.75 

9.545119 
.545524 
.545928 
.546331 
.546735 
.547138 
.517540 
.547943 
.548345 
.548747 

6.74 
6.73 
6.73 
6.72 
6.72 
6.71 
6.71 
6.70 
6.70 
6.69 

0.454881 
.454476 
.454072 
.453669 
.453265 
.452862 
.452460 
.452057 
.451655 
.451253 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 

9.523495 
,523852 
.524208 
.524564 
.524920 
.525275 
.525630 

5.94 
5.94 
5.93 
6.93 
6.92 
5.92 

e  ni 

9.974347 
.974302 
.974257 
.974212 
.974167 
.974122 
.974077 

.75 

.75 
.75 
.75 
.75 
.75 

9.549149 
.549550 
.549951 
.550352 
.550752 
.551  153 
.551552 

6.69 
6.68 
6.63 
6.67 
6.67 
6.67 

0.450851 
.450450 
.450049 
.449648 
.449248 
.448847 
.448448 

30 
29 
23 
27 
26 
25 
24 

37 

.525984 

o.yi 

e  Qrt 

.974032 

75 

.551952 

6.66 

C  fi£ 

.448048 

23 

38 
39 

.526339 
.526693 

o.yu 
5.90 
5.89 

.973987 
.973942 

>5 
.75 

.552351 
.552750 

6.  DO 

6.65 
6.65 

.447649 
.447250 

22 
21 

40 

9.527046 

C  OQ 

9.973897 

9.553149 

A  £/f 

0.446851 

20 

41 
42 
43 

44 

.527400 
.527753 
.528105 
.523458 

VtCfV 

5.88 
5.88 
6.87 

E  Q7 

.973852 
.973807 
.973761 
.973716 

'.75 
.75 
.75 

.553548 
.553946 
.554344 
.554741 

6.64 
6.64 
6.63 
6.63 

.446452 
.446054 
.445656 
.445259 

19 
18 
17 
16 

45 

.528810 

O.O/ 
fi  flft 

.973671 

7A 

.555139 

6.62 

a  £Q 

.444861 

15 

46 
47 

46 
49 

.529161 
.529513 
.529864 
.530215 

o.oo 

5.86 
5.85 
6.85 

5.84 

.973625 
.973580 
.973535 
.973489 

.  /O 

.76 
.76 
.76 
.76 

.555536 
.555933 
.556329 
.556725 

O.O4 

6.61 
6.61 
6.60 
6.60 

.444464 
.441067 
.443671 
.443275 

14 
13 
12 
11 

50 

9.530565 

e  QQ 

9.973444 

9.557121 

0.442879 

10 

51 
5? 
63 
54 
55 
56 
57 
58 
69 
60 

.530915 
.531265 
.531614 
.531963 
.532312 
.532661 
.533009 
.533357 
.533704 
.594052 

O.OO 

5.83 

5.82 
5.82 
5.81 
5.81 
5.80 
5.80 
6.79 
5.79 

.973398 
.973352 
.973307 
.973261 
.973215 
.973169 
.973124 
.973078 
.973032 
.972986 

!76 
.76 
.76 
.76 
.76 
.76 
.76 
.77 
.77 

.557517 
.557913 
.558308 
.558703 
.559097 
.559491 
.559885 
.560279 
,560673 
.561066 

6.59 
6.59 
6.59 
6.58 
6.53 
6.57 
6.57 
6.56 
6.56 
6.55 

.442483 
.442087 
.441692 
.441297 
.440903 
.440509 
.440115 
.439721 
.439327 
.438934 

9 
8 

0 

M. 

Oofllne. 

D.l«. 

Slue. 

D.  1". 

Cotuug. 

D.  1". 

Ikng. 

M. 

100° 


70Q 


-50     .      TABLE  II.   LOGARITHMIC  SINES, 

900                                              160C 

M. 

Sine. 

D.  1". 

Coeine. 

D.  I". 

Taug. 

D.  1". 

Cotang. 

M, 

0 

1 
2 
3 
4 
6 
6 
7 
8 
9 

9.534052 
.534399 
.534745 
.535092 
.535438 
.535783 
.536129 
.536474 
.536818 
.637163 

5.78 
5.78 
5.77 
5.77 
5.76 
5.76 
5.75 
6.75 
6.74 
6.74 

9.972986 
.972940 
.972894 
.972848 
.972802 
.972755 
.972709 
.972663 
.972617 
.972570 

.77 
.77 
.77 
.77 
.77 
.77 
.77 
.77 
.77 
.77 

9.561066 
.561459 
.561851 
.562244 
.562636 
.563(128 
.563419 
.563811 
.564202 
.564593 

6.55 
6.54 
6.54 
6.54 
6.53 
6.53 
6.52 
6.52 
6.51 
6.51 

0.438934 
.438541 
.438149 
.437756 
.437364 
.436972 
.436581 
.436189 
.435798 
.435407 

~60~ 
59 
58 
67 
56 
55 
64 
53 
52 
51 

10 
11 
12 
13 
14 

9.537507 
.537851 
.538194 
.538538 

.538880 

5.73 
5.73 
5.72 
6.71 

9.972524 
.972478 
.972431 
.972385 
.972338 

.77 
.77 
.78 
.78 

9.564983 
.565373 
.565763 
.566153 
.566542 

6.50 
6.50 
6.50 
6.49 

0.435017 
.434627 
.434237 
.433847 
.433458 

60 
49 

48 
47 
46 

15 
16 
17 
18 

.539223 
.539565 
.639907 
.640249 

5.71 
6.70 
5.70 
6.69 

.972291 
.972245 
.972198 
.972151 

.78 
.78 
.78 
.78 

TO 

.566932 
.667320 
.567709 

.568098 

6.49 
6.48 

6.48 
6.47 

H  Af 

.433068 
.432680 
.432291 
.431902 

45 
44 
43 
42 

19 

.640590 

6.68 

.972105 

.7o 
.78 

.668486 

6.47 
6.46 

.431614 

41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.540931 
.541272 
.641613 
.641953 
.542293 
.642632 
.542971 
.543310 
.543649 
.543987 

6.68 
5.67 
6.67 
5.66 
5.66 
6.65 
6.65 
6.64 
664 
6.63 

9.972058 
.97201  1 
.971964 
.971917 
.971870 
.971823 
.971776 
.971729 
.971682 
.971635 

.78 
.78 
.78 
.78 
.78 
.78 
.78 
.79 
.79 
.79 

9.568873 
.569261 
.569648 
.570035 
.570422 
.670809 
.571195 
.671581 
.571967 
.672352 

646 
6.46 
6.45 
6.45 
ti.44 
6.44 
G.43 
6.43 
6.43 
6.42 

0.431127 
.430739 
.430352 
.429965 
.429578 
.429191 
.428806 
.428419 
.428033 
.427648 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 
32 

9.544325 
.544663 
.645000 

6.63 
6.62 

c  co 

9.971588 
.971540 
.971493 

.79 

.79 

9.572738 
.673123 
.673507 

6.42 
6.41 

A  A  1 

0.427262 
.426877 
.426493 

30 
29 

28 

33 
34 
35 
36 
37 
38 

545338 
.545674 
.646011 
646347 
.546683 
.547019 

O.D/ 

6.61 
6.61 
6.60 
6.60 
6.59 

C  en 

.971446 
.971398 
.971351 
.971303 
.971256 
.971208 

>9 
.79 
.79 
.79 
.79 

.673892 
.674276 
.674660 
.676044 
.676427 
.675810 

D.31 

6.40 
6.40 
6.40 
6.39 
6.39 

°d  QU 

.426108 
.425724 
.425340 
.424956 
.424573 
.424190 

27 
26 
26 
24 
23 
22 

39 

.647354 

o.&y 
6.58 

.971161 

'.79 

.676193 

b.JC 
6.88 

.423807 

21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.547689 
.548024 
.548359 
.648693 
.549027 
.649360 
.549693 
.650026 
.650359 
.650692 

6.58 
6.57 
5.57 
6.56 
6.56 
6.55 
6.55 
555 
6.54 
6.54 

9.971113 

.971066 
.971018 
.970970 
.970922 
.970874 
.970827 
.970779 
.970731 
.970683 

.79 

.80 
.80 
.80 
.80 
.80 
.80 
.80 
.80 
.80 

9.676576 

.676959 
.577341 
.677723 
.678104 

.678486 
.578867 
.579248 
.579629 
.680009 

6.37 
6.37 
6.37 
6.36 
6.36 
6.35 
6.35 
6.34 
6.34 
6.34 

0.423424 
.423041 
.422659 
.422277 
.421896 
.421514 
.421  133 
.420752 
.420371 
.419991 

2(1 
19 
18 
17 
16 
16 
14 
13 
12 
11 

6G 

9.551024 

9.970635 

9.680389 

fi  oo 

0.419611 

10 

51 
52 
53 
54 
65 
56 

.551356 
.551687 
.552018 
.552349 
.552680 
.553010 

6.53 
5.53 
652 
552 
5.51 
5.51 

.970586 
.970538 
.970490 
.970442 
.970394 
.970345 

.80 
.80 
.80 
.80 
.80 
.81 

Ol 

.580769 
.681149 
.581528 
.681907 
.682286 
.682665 

DEM 

6.33 
6.32 
6.32 
6.32 
6.31 

£  Ol 

.419231 
.418851 
.418472 
.418093 
.417714 
.417335 

9 

8 
7 
6 
5 
4 

57 
68 
59 
60 

.553341 
.653670 
.654000 
.654329 

6.50 
6.50 
5.49 
6.49 

.970297 
.970249 
.970200 
.970152 

.ol 
.81 
.81 
.81 

.683*44 
.683422 
.583*00 
.584177 

b.dl 
6.30 
6.30 
6.90 

.416956 
.416578 
.416200 
.415823 

3 
2 

1 
0 

M. 

Ooetao. 

D.  1'-. 

ffliift 

D.  1". 

Ootaiig. 

D.  1". 

Tang. 

M. 

1100 


60° 


COSINES,    TANGENTS,    AND    COTANGENTS. 


a  io 


M. 

Slue. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

1 
2 
3 
4 
5 

9.554329 
.554658 
.664US7 
.655315 
.55r,643 
.655971 

5.48 

5.43 
5.47 
6.47 
6.46 

9.970152 
.970103 
.970055 
.970006 
.969957 
.969909 

.81 
.81 
.81 
.81 
81 

Q  1 

9.584177 

.584555 
.584932 
.585309 
.585686 
.586062 

6.29 
6.29 
6.28 
6.28 
6.28 

0.415823 
.415445 
.415068 
.414691 
.414314 
.413938 

60 
69 
68 
67 
56 
55 

6 

7 
8 

.556299 
.556626 
556953 

5.46 
5.45 
5.45 

.969860 
.969P11 
.969762 

•  01 

.81 

.81 

Ql 

.586439 
.5868  1  5 
.587190 

6.27 
6.27 
6.26 

.413561 
.413185 
.412810 

54 
53 
52 

9 

.557280 

5.44 
6.44 

.969714 

,ol 
.81 

.587566 

6.26 
6.26 

.412434 

61 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9  557606 
.557932 
.558258 
.55.3583 
.558909 
.559234 
.559558 
.559883 
.560207 
.560531 

5.44 

5.43 
5.43 
5.42 
5.42 
5.41 
5.41 
6.40 
5.40 
6.39 

9.969665 
.%9616 
.969567 
.969518 
.969469 
.969420 
.969370 
.969321 
.969272 
.969223 

.82 
.82 
.82 
.82 
82 
.82 
.82 
.82 
.82 
.82 

9.587941 
.588316 
.588691 
.589066 
.589440 
.589814 
.590188 
.590562 
.590935 
.591308 

6.25 
6.25 
6.24 
6.24 
6.24 
6.23 
6.23 
6.22 
6.22 
6.22 

0.412059 
.411634 
.411309 
.410934 
.410560 
.410186 
.409812 
.409438 
.409065 
.408692 

60 
49 

48 
47 
46 
45 
44 
43 
42 
41 

20 

9.560855 

9.969173 

9.591681 

0.408319 

40 

21 

.561178 

5.39 

.969124 

.82 

QO 

.592054 

6.21 

.407946 

39 

22 
23 

.561501 
.561824 

6.38 
5.38 

.969075 
.969025 

•CM 

.82 

.592426 
.592799 

6.21 
6.20 

.407574 
.407201 

38 
37 

24 

25 

.562146 
.562468- 

5.37 
6.37 

.968976 
.968926 

'.83 

.593171 
.593542 

6.20 
6.20 

.406829 
.406458 

36 
35 

26 
27 
28 
29 

.562790 
.563112 
.563433 
.563755 

5.37 
6.36 
6.36 
6.35 
5.35 

.968377 
.968827 
.968777 
.968728 

.83 
.83 
.83 
.83 
83 

.593914 
.594285 
.594656 
.595027 

6.19 
6.19 
6.18 
6.J8 
6.18 

.406086 
.405715 
.405344 
.404973 

34 
33 
32 
31 

30 
31 
32 

9.564075 
.5643% 
.564716 

5.34 
5.34 

9.96S678 
.968623 
.968573 

.83 
83 

9.595393 
.595763 
.596138 

6.17 
6.17 

0.404602 
.404232 
.403862 

30 
29 

28 

33 
34 
35 

.565036 
.565356 
.565676 

5.33 
6.33 
5.32 

.968523 
.968479 
.968429 

.83 
.83 
.83 

.596508 
.596878 
.597247 

6.16 
6.16 
6.16 

.403492 
.403122 
.402753 

27 
26 
25 

36 
37 
38 
39 

.565995 
.566314 
.566632 
.566951 

5.32 
6.32 
6.31 
6.31 
6.30 

.968379 
.968329 
.968278 
.968228 

.83 
.83 
.83 

.84 
.84 

.597616 
.597985 
.598354 
.698722 

6.15 
6.15 
6.15 
6.14 
6.14 

.402384 
.402015 
.401646 
.401278 

24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
48 
47 
48 
49 

9.567269 
.667587 
.567904 
.568222 
.568539 
.568856 
.569172 
.569488 
.569304 
.670120 

5.30 
6.29 
5.29 
6.28 
6.28 
6.28 
6.27 
5.27 
6.26 
6.26 

9.968178 
.968123 
.968078 
968027 
.967977 
.967927 
.967876 
.967826 
.967775 
.967725 

84 
.84 
.84 
.84 
.84 
-  .64 
.81 
.84 
.84 
.84 

9.599091 
.599459 
.599827 
.600194 
.600562 
.600929 
.601296 
.601663 
.602029 
.602395 

6.13 
613 
6.13 
6.12 
6.12 
6.12 
6.11 
6.11 
6.10 
6.10 

0.400909 
.400541 
.400173 
.399806 
.399438 
.399071 
.398704 
.398337 
.397971 
.397605 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 

9.570435 
.570751 

5.25 

9.967674 
.967624 

.84 

9.602761 
.603127 

6.10 

0.397239 
.396873 

10 
9 

52 

.671066 

6.25 

.967573 

.^4 

.603493 

6.09 

.396507 

8 

53 
54 
55 

.571380 
.671695 
:  572009 

5.24 
6.24 
5.24 

.967522 
.%7471 
.967421 

.85 
.85 
.85 

.603858 
.604223 
.604588 

6.09 
6.09 
6.08 

.396142 
.395777 
.395412 

7 
6 
6 

56 
57 

53 

.572323 
672636 

572950 

6.23 
6.23 
6.22 

.967370 
.967319 
.967268 

.85 

.85 
.85 

.604953 
.605317 
.605682 

6.08 
6.07 
6.07 

.395047 
.394683 
.394318 

4 
3 
2 

59 
60 

.673263 
.673575 

6.22 
6.21 

.967217 
.967166 

.85 
.85 

.6fl6046 
.606410 

6.07 
6.06 

.393954 
.393590 

0 

M. 

Oosino. 

D.l". 

Sine. 

D.  1". 

Gotang. 

"D^ 

Tang. 

M. 

68° 


252            TABLE  II.   LOGARITHMIC  SINES, 

330                               isr 

M 

Bine. 

D.  1".   CoPine. 

D.I". 

Taiig. 

D.  1". 

Cotang. 

M. 

0 

1 

9573575 

.673888 

5.21 

9.967166 
.967115 

.85 

QC 

9.606410 
.606773 

6.00 

c  rt/s 

0.393590 
.393227 

60 
59 

2 
3 

4 
5 
6 

.574200 
.574512 
.574824 
.575136 
.675447 

5.2( 
5.2( 
5.20 
5.19 
5.19 

.967064 
.967013 
.966961 
.966910 
.966859 

,OO 

.85 
.85 
.85 

.85 

.607137 
.607500 
.607863 
.608225 

.60arx38 

o.Uo 
6.05 
6.05 
6.05 
6.04 

.392863 
.392500 
.392137 
.391775 
.391412 

58 
57 
56 
56 
54 

7 

.575758 

5.18 

.966808 

.86 

.608950 

6.04 

£  no 

.391050 

63 

8 
9 

.576069 
.576379 

5.  18 
5.17 
5.17 

.966756 
.966705 

'M 

.86 

.609312 
.609674 

b.Uo 
6.03 
6.03 

.390688 
.390326 

61 

10 
11 

9.576889 
.576999 

5.17 

9.966653 
.966602 

.86 

9.610036 
.610397 

6.02 

c  no 

0.389964 
.389603 

50 
49 

12 
13 

.577309 
.577618 

£.16 
5.16 

.966550 
.966499 

.86 

.86 

.610759 
.611120 

o.uz 
6.02 
*•  ni 

.389241 

.383880 

48 
47 

14 

.577927 

5.15 

.966447 

.86 

.611480 

O.U1 

.388520 

46 

15 
16 

.578236 
.578545 

5.15 
5.14 

.966395 
966344 

.86 
.86 

611841 

.612201 

6.01 
6.01 

£  nn 

.388159 
.387799 

45 
44 

17 

.678853 

5.14 

966292 

.86 

612561 

O.IKJ 

.387439 

43 

18 
19 

579162 
.679470 

5.14 
5.13 
5.13 

966240 
.966188 

.86 
.86 
.86 

612921 
.€13281 

6.00 
6.00 
5.99 

.387079 
.386719 

42 
41 

20 
21 

9.579777 

.680035 

'5.12 

9.966136 

.966035 

.87 

9.613541 
.614000 

5.99 

0.336359 
.386000 

40 
39 

22 
23 

.580392 
.580699 

5.12 
5.11 

.966033 
.965981 

.87 
.87 

.614359 
.614718 

5.98 

•   d  QQ 

.335641 

.385282 

38 
37 

24 
25 
26 
27 

.581005 
.581312 
.581618 
.531924 

5.  1  1 
511 
5.10 
5.10 

.965929 
.965376 
.965324 
.965772 

.87 
.87 
.87 
.87 

.615077 
.615435 
.615793 
.616151 

5.9o 
6.97 
5.97 
6.97 

.334923 
.384565 
.384207 
.383849 

36 

35 
34 
33 

28 
29 

.582229 
.532535 

5.09 
5.09 
5.09 

.965720 
.965663 

.87 
.87 
.87 

.616509 
.616867 

5.96 
6.96 
5.96 

.383491 
.383133 

32 
31 

30 
31 
32 
33 
34 

9.582840 
.583145 
.583449 
.583754 
.584058 

5.08 
5.08 
5.07 
5.07 

9.965615 
.965563 
.965511 
.965458 
.965406 

.87 
.87 
.87 
.87 

9.617224 
.617582 
.617939 
.61  8295 
.613652 

5.95 
5.95 
5.95 
5.94 

0.382776 
.382418 
.382061 
.381705 
.381348 

30 

29 

28 
27 
26 

35 

.584361 

5.06 

.965a53 

.88 

.619008 

5.94 

.380992 

26 

36 
37 
33 

.584665 
.534968 
.535272 

5.06 
5.06 
5.05 

.965301 
.965248 
.965195 

.88 
.88 
.88 

.619364 
.619720 
.620076 

5.94 
5.93 
5.93 

.380636 
.380280 
.379U24 

24 
23 

22 

39 

.585574 

6.05 
5.04 

.965143 

.88 
.88 

.620432 

6.93 
5.92 

.379568 

21 

40 

9.585377 

9.965090 

9.620787 

0.379213 

20 

41 

.586179 

5.04 

.965037 

.88 

.621  142 

5.92 

378858 

19 

42 

536482 

5.04 

.964984 

.88 

.621497 

5.92 

r  ni 

.378503 

18 

43 

536783 

5.03 

.964931 

.88 

.621652 

5.91 

378148 

17 

44 
45 
46 

.537085 
587336 

.587638 

5.03 
5.02 

5.02 

.964379 
.964326 

.964773 

.88 
.88 
.83 

.622207 
.622561 
.622915 

5.91 
6.91 
5.90 

.377793 
.377439 
377085 

16 
15 
14 

47 

.587939 

5.01 

.964720 

.88 

.623269 

5.90 

.376731 

13 

48 

.583289 

5.01 

.964666 

.88 

623623 

5.90 

n  QQ 

376377 

12 

49 

.588590 

5.01 
5.00 

.964613 

.89 
.89 

.623976 

5.H9 
5.89 

.376024 

11 

50 

9.58,3890 

9.964560 

9.624330 

0.375670 

10 

51 

.589190 

5.00 

.964507 

.89 

.624633 

C.  QQ 

.375317 

9 

52 
53 
54 
45 

539439 
589789 
.590088 
.590387 

4.99 
4.99 
4.99 
4.93 

.964454 
.964400 
.964347 
.964294 

.89 
.89 
.89 
.89 

.625036 
.625388 
.625741 
.626093 

o.bo 
5.88 
5.88 
5.87 

C  QT 

.374964 
.374612 
.374259 
.373907 

8 
7 
6 
5 

56 

.590686 

4.98 

.964240 

.89 

.626445 

o.o/ 

.373555 

4 

67 

58 

.590984 
.591282 

4.97 
4.97 

.964187 
.964133 

.89 
.89 

.626797 
.627149 

5.87 
5.86 

C  00 

.373203 
.372851 

3 
2 

59 

.591530 

4.97 

.964080 

.89 

.627501 

o.oo 

.372499 

1 

60 

.591878 

4.96 

.964026 

.89 

.627852 

5.86 

.372148 

0 

M. 

Oodne. 

D.  1". 

Sine. 

D.  1". 

Cotaug. 

D.I". 

Tang. 

M. 

COSINES,  TANGENTS,  AND  COTANGENTS.         253 

380                                          166° 

H. 

Bine. 

D.  1". 

Cosine. 

D.  1" 

T«ng. 

D.  1". 

Cotang. 

M. 

9.691878 

9  96402o 

QQ 

9.627852 

0.372148 

60 

.692176 

1  <r 

!  963972 

.OS! 

89 

.628203 

5.85 

e  QC 

.371797 

59 

7 

.592473 
.692770 
.593'  67 
.593363 
.693659 
.693955 

1*95 

4.95 
4.94 
4.94 
493 

.963919 
:963865 
.96381  1 
.963757 
.963704 
.963650 

^90 
.90 
.90 
.90 

.628554 
.628905 
.629255 
.629606 
.629956 
.630306 

O.OO 

6.85 
5.84 
5.84 
6.84 
5.83 

r  QO 

.371446 
.371095 
.370745 
.370394 
.370044 
.369694 

68 
57 
56 
55 
54 
53 

8 

.694251 

00 

.963596 

'on 

.630656 

O.OO 
C  QO 

.369344 

52 

9 

.594547 

4^92 

.963542 

!90 

.631005 

o.oo 

6.82 

.368995 

51 

10 
11 
12 
13 
14 
IG 
16 
17 

9.594842 
.595137 
.595432 
.595727 
.596021 
.596315 
.596609 
.696903 

4.92 
4.91 
4.91 
4.91 
4.90 
4.90 
4.89 

A  QQ 

9.963488 
.963434 
.963379 
.963325 
.963271 
.903217 
.963163 
.963108 

.90 
.90 
.90 
.90 
.90 
.90 
.91 
at 

9.631355 
.631704 
.632053 
.632402 
.632750 
.633099 
.633447 
.633795 

5.82 
6.82 
6.81 
5.81 
5.81 
5.80 
6.80 

c  on 

0.368645 
.368296 
.367947 
.367598 
.367250 
.366901 
.366553 
.366205 

60 
49 

48 
47 
46 
45 
44 
43 

18 
19 

.697196 
.697490 

4.  by 
4.89 
4.88 

.963054 
.962999 

.VI 

.91 
.91 

.634143 
.634490 

5.UU 
6.79 
6.79 

.365857 
.365510 

42 
41 

20 
21 
22 
23 
24 
25 
20 
27 
28 
29 

9.597783 
.698075 
.698363 
.59866C 
.698952 
.699244 
.699536 
.599827 
.6001  18 
.600409 

4.88 
4.88 
4.87 
4.87 
4.86 
4.86 
4.86 
4.85 
4.85 
4.84 

9.962945 
.962890 
.962336 
.962781 
.962727 
.962672 
.962617 
.962562 
.962508 
.962453 

.91 
.91 
.91 
.91 
.91 
.91 
.91 
.91 
.91 
.92 

9.634838 
.635185 
.635532 
.635879 
.636226 
.636572 
.636919 
.637265 
.637611 
.637956 

6.79 
6.78 
6.78 
5.78 
6.78 
6.77 
6.77 
5.77 
6.76 
6.76 

0.365162 
.364S15 
.364468 
.364121 
363774 
.363428 
.363081 
.362735 
.362389 
.362044 

40 
39 
38 
37. 
36 
35 
34 
33 
32 
31 

ao 

31 
32 
33 
34 

35 
36 
37 
38 

9.600700 
.600990 
.601280 
.601570 
.601860 
.602150 
.602439 
.602728 
.603017 

4.84 
4.84 
4.83 
4.83 
4.83 
4.82 
4.82 
4.81 

A  Qt 

9.962398 
.962343 
.902288 
.962233 
.962178 
.962123 
.962067 
.962012 
.961957 

.92 
.92 
.92 
.92 
.92 
.92 
.92 
.92 

9.638302 
.638647 
.638992 
.639337 
.639682 
.640027 
.640371 
.640716 
.641060 

6.76 
6.75 
6.75 
5.75 
5.74 
6.74 
5.74 
6.73 

0.361698 
.361353 
.361008 
.360663 
.360318 
.359973 
.359629 
.359284 
.358940 

30 
29 
23 
27 
26 
25 
24 
23 
22 

39 

.603305 

4.ol 

4.81 

.961902 

.92 
.92 

.641404 

5.73 
6.73 

.358596 

21 

40 
41 
42 

9.603594 
.603882 
.604170 

4.80 
4.80 

A  7Q 

9.961846 
.961791 
.961735 

.92 
.92 

no 

9.641747 
.642091 
.642434 

6.73 
5.72 

e  TO 

0.358253 
.357909 
.357566 

20 
19 
18 

43 
14 
45 
46 

47 

.604457 
.604745 
605032 
.605319 
.605606 

*±.  /y 
4.79 
4.79 
4.78 
4.78 

A  7<D 

.961680 
.961624 
.961569 
.961513 
.961458 

.9.6 

.93 
.93 
.93 

.93 

.642777 
.643120 
.643463 
.643806 
.644148 

9.  f  * 

5.72 
6.71 
6.71 
5.71 

c  7A 

.357223 
.356880 
.356537 
.356194 
.355852 

17 
16 
15 
14 
13 

48 
49 

.605892 
.606179 

4.  /O 

4.77 
4.77 

.961402 
.961346 

!93 
.93 

.644490 
.644832 

O./U 

6.70 
6.70 

.355510 
.355168 

12 
11 

50 

9.6064«5 

9.961290 

9.645174 

e  ca 

0.354826 

10 

61 

.606751 

4.76 

.961235 

.93 

.645515 

5.  69 

E  CO 

.354484 

9 

52 
53 
54 
65 

.607036 
.607322 
.607607 
.607892 

4.76 
4.75 
4.75 

.961179 
.961123 
961067 
.961011 

!93 
.93 
.93 

.645^57 
.646199 
.616540 
646881 

o.oy 
d.69 
5.69 
5.68 

K  CU 

.354143 
.353801 
.353460 
.353119 

8 
7 
6 
6 

56 
67 

.608177 
.60>461 

4.74 
4.74 

.960955 
.960S99 

.93 
.93 

647222 
.647562 

O.DO 

5.68 

.352778 
.352438 

4 
3 

68 
69 

60 

.608745 
.609029 
.609313 

4.74 
473 
4.73 

.960843 
.960786 
.960730 

.94 
.94 
.94 

.647903 
.64S243 
.648583 

5.67 
5.67 
5.67 

.352097 
.351757 
.351417 

2 
0 

M. 

Cosine. 

D.  1". 

Sine 

D.  1". 

Co  tang 

D.I". 

Timg. 

M." 

1130 


'254 


TABLE    II.       LOGARITHMIC    SINES, 


156> 


M. 

Sine. 

D.l". 

Codue. 

D.  1". 

Tang. 

D.I'. 

Cotang. 

M. 

1 
2 
3 
4 
6 
6 
7 
8 
9 

9609313 

.609.397 
.609380 
.610164 
.610447 
.610729 
.611012 
611294 
.61  1576 
.611868 

4.73 
4.72 
4.72 
4.72 
4.71 
4.71 
4.71 
4.70 
4.70 
4.69 

9.960730 
.960674 
.960618 
.960561 
.960505 
.96(448 
.960392 
.960335 
.960279 
.960222 

.94 
.94 
.94 
.94 
.94 
.94 
.94 
.94 
.94 
.94 

9.648583 
.64-1923 
.649263 
.649602 
.649942 
.651  (231 
.650620 
.650959 
.651297 
.651636 

6.67 
6.66 
5.66 
6.66 
665 
6.66 
5.65 
6.64 
5.64 
5.64 

0.351417 
.351077 
.350737 
.350398 
.350058 
.349719 
.349380 
349041 
.348703 
.348364 

59 
58 
67 
56 
55 
64 
63 
52 
51 

10 

9.612140 

9.960165 

9.651974 

K.  G.A 

0.343026 

60 

11 

.612421 

A   £Q 

.960109 

.95 

.652312 

6.64 

.34763.3 

49 

12 
13 
14 
15 
16 
17 
18 

.612702 
.612933 
.613264 
613545 
.613825 
.614105 
.614335 

4.  by 
4.63 
4.63 
4.63 
4.67 
4.67 
4.67 

j    CO 

.96)062 
.959995 
.959933 
.959382 
.959325 
.959763 
.959711 

.95 
.95 
.95 
.95 
.95 
.95 
.95 

O' 

.652650 
.652933 
.6;53326 
.653663 
.654000 
.654337 
.654674 

5.63 
6.63 
6.63 
662 
6.62 
5.62 
6.62 

C    ,•  I 

.347350 
.347012 
.346674 
.346337 
.346000 
345663 
.345326 

48 
47 
46 
46 
44 
43 
42 

19 

.614665 

4.  DO 

4.66 

.959654 

.»o 
.96 

.655011 

O.Dl 

6.61 

.344989 

41 

20 
21 
22 
23 
24 
26 
26 

9.614944 
615223 
.615502 
.615781 
616060 
616333 
.616616 

4.65 
4.65 
4.65 
4.64 
4.64 
4.64 

1    £O 

9  959596 
.9.59539 
.959482 
.9f>9425 
.9.59368 
.959310 
.959263 

.95 

.95 
.95 
.95 
.96 
.96 

9.655348 
.655634 
656020 
.666356 
.6.56692 
.657028 
.6.57364 

6.61 
6.61 
6.60 
6.60 
6.60 
6.59 

t  CO 

0.344652 
344316 
.343930 
.343644 

.343303 
.342972 
.34*636 

40 
39 
38 
37 
36 
35 
34 

27 
28 
29 

.616394 
.617172 
.617450 

4.  DO 

4.63 
463 
4.62 

.959195 
.9.59133 
.959080 

.96 
96 
.96 
.98 

.657699 
.658034 
.658369 

O.5B 
6.59 
6.58 
6.68 

.342301 
.341966 
.341631 

33 
32 
31 

30 

9.617727 

9.959023 

9.658704 

e  eo 

0.341296 

30 

31 

.618004 

4.62 

.953965 

.96 

.659039 

5.  5o 

.340%! 

29 

32 
33 

36 

36 
37 
38 
39 

.618231 
.618553 
.618334 
.619110 
619336 
.619662 
.619933 
.620213 

4.61 
4.61 
4.61 
4.60 
4.60 
4.60 
4.69 
4.59 
4.59 

.953903 
.9598(0 

.953792 
.958734 
.953677 
.953619 
.953561 
.958503 

.96 
.96 
.96 
.96 
.96 
.96 
.97 
.97 
.97 

659373 
659708 
660042 
660376 
660710 
661043 
.661377 
.661710 

6.58 
6.57 
6.67 
6.67 
5.56 
6.56 
656 
6.56 
6.65 

.340627 
.340292 

.339624 
.339290 
.333957 
.33^623 
.338290 

28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 

9.620488 
.620763 
.621033 
.621313 
.621537 
.621361 
.622135 
.622409 
.622632 

4.53 
4.53 
4.58 
4.57 
4.57 
4.57 
4.56 
4.56 

9.958445 
.953337 
.953329 
.953271 
.958213 
.958154 
.953096 
.953033 
.957979 

.97 
.97 
.97 
.97 
.97 
.97 
.97 
.97 

9662043 
662376 
.662709 
.663042 
.663375 
.663707 
664039 
.664371 
664703 

6.55 
655 
6.54 
654 
654 
6.54 
653 
5.53 

0.337957 
.337G24 
337291 
.336958 
336625 
336293 
335961 
33:5629 
335297 

20 
19 
18 
17 
16 
16 
14 
13 
12 

49 

.622956 

4.56 
4.55 

.957921 

.97 
.97 

665035 

5.53 
653 

334965 

11 

60 

9.623229 

9.957863 

9.665366 

C    CO 

0.334634 

10 

51 

.623502 

4.55 

957804 

.97 

665698 

6  52 

334302 

9 

52 
63 

.623774 
.624047 

4.54 
4.54 

.957746 

957687 

.98 
.98 

.666029 
666360 

5.52 
5.52 

.333971 
.333640 

8 
7 

64 

.624319 

4.54 

957628 

.98 

666691 

5.61 

.333309 

6 

55 

.624591 

4.53 

.957570 

.98 

667021 

5.51 

332979 

5 

66 

.624363 

4.53 

.95751  1 

.93 

.667352 

5.51 

.332648 

4 

57 
68 
69 

.625135 
.625106 
625677 

4.53 
4.52 
4.52 

.9-57452 
9H7393 
.957335 

.98 
.98 
.93 

667682 
668013 
.663343 

5.51 
5.50 
6.60 

e  cfi 

.332318 

.331987 
.331657 

3 
2 

1 

60 

.625948 

4.52 

.957276 

.98 

.663673 

O.oU 

.331327 

0 

M. 

Coetaa. 

D.  1". 

Slue. 

D.  1". 

Ootang. 

D.  1".       Ifcng. 

if- 

COSINES,    TANGENTS,    AND    COTANGENTS. 


164° 


M 

Blue. 

D.l". 

Cosine. 

D  1" 

Tiuig. 

D.l". 

Ootaug. 

M. 

0 
1 

i 

9.625948 
.626219 
.626490 

4.61 
461 

9.957276 
.957217 
.957168 

.98 
.98 

9.66S673 
.669002 
.669332 

6.60 
6.49 

0.331327 
.330998 
.330668 

60 
69 
68 

3 

.626760 

4.61 

.957099 

flN 

.669661 

fila 

.330339 

67 

4 

6 
6 

7 
8 
9 

.627030 
.627300 
.627670 
.627840 
.628109 
.628378 

i.K 
460 
4.19 
4.49 
4.49 
4.48 

.967040 
.956981 
.956921 
.956662 
.956803 
.956744 

'99 
.99 
.99 
.99 
.99 
99 

.669991 
.670320 
.670649 
.670977 
.671306 
.671636 

6^49 
6.48 
6.48 
6.48 
6.47 
6.47 

.330009 
.329680 
.329351 
.329023 
.328694 
.328365 

66 
65 
64 
63 
62 
61 

ic 

11 

9.628647 
.628916 

4.48 

9.956684 
.956625 

.99 

oo 

9.671963 

.672291 

6.47 

0.328037 
.327709 

60 
49 

IS 

.629185 

4.48 

.956566 

.yy 

.672619 

6.47 

.327381 

48 

13 
14 

.629453 
.629721 

4.47 
4.47 

.956506 
.956447 

.99 
.99 

.672947 
.673274 

6.46 
6.46 

C     4£ 

.327053 
.326726 

47 
46 

16 
16 

.629989 
.630257 

446 

A    Ati 

.956387 
.956327 

'99 

.673602 
.673929 

O.48 

6.46 

.326398 
.326071 

46 
44 

17 
18 
19 

.630524 
.630792 
.631059 

4.45 
446 
4.45 
4.45 

956268 
.956208 
.966148 

.99 
.99 
1.00 
1.00 

.674257 
.674584 
.674911 

6.46 
6.46 
6.46 
6.46 

.325743 
.325416 
.325089 

43 
42 
41 

» 

22 
13 
24 
*6 
26 
27 
28 
29 

9.631326 

.631593 
.631859 
.632  25 
.632392 
.632658 
.632923 
.633189 
.633454 
.633719 

4.46 
4.44 

444 
4.44 
443 
443 
443 
442 
442 
4.42 

9.956089 
.956029 
.955969 
.955909 
.955S49 
.955789 
.955729 
.955669 
.955609 
.955648 

l.CO 
1.00 

1.00 
1.00 
1.00 
1  00 
1.00 
100 
100 
1.00 

9.675237 
.6.~5564 
.676890 
.676217 
.676543 
.676869 
.677194 
.677620 
.677846 
.678171 

6.44 
6.44 
6.44 
6.44 
6.43 
643 
643 
642 
6.42 
6.42 

0.324763 
.324436 
.324110 
.323783 
.323457 
.323131 
.322806 
.322480 
.322154 
321629 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 

9.633984 
.634249 
.634514 
.634r8 
.635042 
.635306 
.6355^0 

4.41 
441 
4.41 
4.40 
4.40 
4.40 

A  on 

9.955488 
.955428 
.955368 
.955307 
.955247 
.955186 
.955126 

1.00 
1.01 
1.01 
1.01 
1.01 
1.01 

1   II] 

9.678496 
.678821 
.679146 
.679471 
.679795 
.680120 
.680444 

642 
6.41 
6.41 
6.41 
5.41 
6.40 

0.321504 
.321179 

.320854 
.320529 
.320205 
.319880 
.319556 

30 
29 
28 
27 
26 
25 
24 

37 

38 
39 

.635834 
.636097 
.636360 

4.o9 
4.39 
4.39 
4.38 

.955(*5 
.955005 
.954944 

I.UI 

1.01 
1.01 
1.01 

.680768 
.681092 
.681416 

6.40 
6.40 
6.40 
6.39 

.319232 
.316908 
.318584 

23 
22 
21 

40 
41 
42 

9.636623 

.6368S6 
.637148 

4.38 
4.38 

A    *¥* 

9.954883 
.954823 
.954762 

1.01 
1.01 

9.681740 

.682063 
.682387 

6.39 
5.39 

c  on 

0.316260 
.317937 
.317613 

20 
19 

18 

43 
44 

.637411 
.637673 

4.  J7 
4.37 

A  or 

.954701 
.954640 

1.01 
101 

.682710 
.683033 

6.  oil 
5.38 

e  oo 

.317290 
.316967 

17 
16 

45 

46 
47 

48 
49 

.637935 
.638197 
.638458 
.638720 
.638981 

4.o7 
4.36 
4.36 
4.36 
4.35 
4.35 

.954579 
.954518 
.954457 
.954396 
.954335 

L02 
1.02 
1.02 
1.02 
1.02 

.683356 
.683679 
.684001 
.684324 
.684646 

D.OO 

6.38 
5.38 
6.37 
6.37 
6.37 

.316644 
.316321 
315999 
.315676 
.315354 

15 
14 
13 
12 
11 

60 
61 
62 
53 

9.639242 
.639503 
.639764 
.640024 

4.35 
4.34 
4.34 

9.954274 
.954213 
.954152 
.954090 

1.02 
1.02 
1.02 

9.684968 
.685290 
.685612 
.685934 

6.37 
6.36 
6.36 

0.315032 
.314710 
.314388 
.314066 

1C 
9 

8 
7 

64 

.640284 

4.34 
400 

.954029 

.02 

.686255 

5.36 

.313745 

6 

65 

.640544 

.00 

A    9Q 

.953968 

.02 

.686577 

6.36 

e  oe 

.313423 

6 

56 

67 
68 

.640804 
.641064 
.641324 

4.OO 
433 
4.32 

.953906 
.953845 
.953783 

.02 
.02 
.03 

.6S6898 
.687219 
687540 

O.OO 

6.35 
6.35 

.313102 
.312761 
.312460 

4 

3 
2 

69 
60 

.641583 
.641842 

4.32 
4.32 

.953722 
.SC3660 

.03 
1.03 

687861 

.638182 

6.35 
6.35 

.312139 
.311818 

I 
0 

M 

Cosine 

D  1" 

Slue 

D.  1". 

Cotaag- 

D.I'. 

TMg. 

"if 

TABLE    II.       LOGARITHMIC    SINES, 


460 


1680 


M. 

Sine.  1 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotwg. 

M. 

~0 
1 
2 
3 

4 
5 
6 

7 

9.64184V 
.642101 
.642360 
.642613 
.642877 
.643135 
.643393 
.643650 

4.32 
4.31 
4.31 
4.31 
4.30 
4.30 
4.30 

9.963660 
.953599 
.953537 
.953475 
.953413 
.953352 
.953290 
.953228 

1.03 
1.03 
1.03 
1.03 
1.03 
1.03 
1.03 

9.688182 
.688502 
.688823 
.689143 
.689463 
.689783 
.690103 
.690423 

5.34 
5.34 
5.34 
5.34 
5.33 
6.33 
6.33 

K  QO 

0.311818 

.31149? 
.311177 
.310857 
.310537 
.310217 
.309897 
.309577 

60 
59 
68 
67 
66 
66 
64 
63 

8 
9 

.643908 
.644165 

4^29 
4.29 

.953166 
.953104 

1.03 
1.03 

.690742 
.691062 

D.OO 

6.32 
6.32 

309258 
.308938 

62 
61 

10 
11 

9.644423 

.644680 

4.28 

4  OQ 

9.953042 
.952980 

1.03 

1  1  Li 

9.691381 

.691700 

6.32 

e  oo 

0.308619 
.308300 

60 
49 

12 
13 
14 
15 
16 
17 
18 
19 

.644936 
.645193 
.645450 
.645706 
.645962 
.646213 
.646474 
.646729 

4.-6O 

4.28 
4.27 
4.27 
4.27 
4.26 
4.26 
4.26 
4.26 

.952918 
.952855 
.952793 
.952731 
.952669 
.952606 
.952544 
.952481 

1  .'rl 

1.04 
1.04 
1.04 
1.04 
1.04 
1.04 
1.04 
1.04 

.692019 
.692338 
.692656 
.692975 
.693293 
.693612 
.693930 
.694248 

O.'fi 

5.31 
6.31 
6.31 
6.31 
5.30 
6.30 
6.30 
6.30 

.307981 
307662 
&  (7344 
.307025 
.306707 
.306388 
.306070 
.306752 

48 
47 
16 
45 
14 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.646984 
.647240 
.647494 
.647749 
.648004 
.648258 
648512 
.648766 
.649020 
.649274 

4.25 
4.25 
4.25 
4.24 
4.24 
4.24 
4.23 
4.23 
4.23 
4.22 

9.952419 
.952356 
.952294 
.952231 
.952168 
.952106 
.952043 
.951980 
.951917 
.951854 

1.04 
1.04 
1.04 
1.04 
1.05 
1.05 
1.05 
1.05 
105 
1.05 

9.694566 
.694,833 
.695201 
.695518 
.695836 
.696153 
.696470 
.696787 
.697103 
.697420 

6.29 
6.29 
6.29 
6.29 
6.29 
6.28 
6.28 
6.29 
5.23 
6.27 

0.305434 

.305117 
.304799 
.304482 
.304164 
.303847 
303530 
.303213 
.302897 
.302530 

40 
39 
33 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.649527 
.649781 
.650031 
.650287 
.650539 
.650792 
.651044 
.651297 
.651549 
.651800 

4.22 
4.22 
4.22 
4.21 
4.21 
4.21 
4.20 
4.20 
420 
4.19 

9.951791 
.951728 
.951665 
.951602 
.951539 
.951476 
.951412 
.951349 
.951286 
.951222 

1.05 
1.05 
1.05 
1.05 
1.05 
1.05 
1.05 
1.06 
1.06 
1.06 

9.697736 
.698053 
.698369 
.698685 
.699001 
.699316 
.699632 
.699947 
.700263 
.700578 

6.27 
6.27 
6.27 
6.26 
6.26 
6.2ft 
6.26 
6.26 
5.25 
6.25 

0.302264 

.301947 
.301631 
.301316 
.300999 
.300684 
.300363 
.300053 
.299737 
.299422 

30 
99 
98 
97 
96 
96 
24 
93 
22 
21 

40 
41 
42 
43 

9.652052 
.652304 
.652555 
.652806 

4.19 
4.19 

.18 

9.951159 

.951096 
.951032 
.950968 

1.06 
1.06 
1.06 

9.700893 
.701208 
.701523 

.701837 

6.25 
6.25 
6.24 

0.299107 
.298792 
.298477 
.298163 

90 
19 
18 
17 

44 

.653057 

.18 

.950905 

1.06 

1f\£t 

.702152 

6.24 

.297848 

16 

45 
46 

.653308 
.653558 

.18 
.18 

.950841 
.950778 

.Uo 
1.06 

1  HA 

.702466 
.702781 

6.24 
6.24 

e  o>4 

.297534 
.297219 

16 
14 

47 

.653808 

.17 

.950714 

J.Ub 
i  rus 

.703095 

5.4* 
e  «io 

.296905 

13 

48 
49 

654059 
654309 

.17 
.17 
.16 

.950650 
.950586 

l.Uo 
1.06 
1.06 

.703*^9 
.703722 

5.23 

6.23 
6.23 

.296591 
.296278 

19 
11 

50 
51 
52 
53 
54 
55 
56 
57 

9.654558 
.654808 
.655053 
.655307 
.655556 
.655805 
.656054 
.656302 

.16 
.16 
.15 
.15 
.15 
.15 
.14 

9.950522 
.950453 
.950394 
.950330 
.950266 
.950202 
.950138 
.950074 

1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
1.07 

9.704036 
.704350 
.704663 
.704976 
.705290 
.705603 
.705916 
.706228 

6.23 
6.22 
6.22 
6.22 
5.22 
5.22 
5.21 

0.295964 
.295650 
.295337 
.295024 
.994710 
.294397 
.294084 
.293772 

10 
9 
8 
7 
6 
6 
4 
3 

58 
59 
60 

.656551 
.656799 
.657047 

4.14 
4.14 
4.13 

.950010 
.949945 
.949381 

1.07 
1.07 
1.07 

.706541 
.706854 
.707166 

5.21 
6.21 
6.21 

.293459 
.293140 
.292834 

9 

0 

M. 

Cosine. 

D.  1". 

Bine. 

D.  1". 

Ootang. 

D.1". 

Tang 

M. 

COSINES,    TANGENTS,    AND    COTANGENTS. 


15* 


M. 

Slue. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Co  tang. 

M. 

0 

9.657047 

9.949831 

fY7 

9.707166 

0.292834 

60 

1 
2 
3 
4 
6 
6 
7 
5 

.657295 
.657542 
.657790 
.658037 
.658234 
.658531 
.658778 
659025 

4.13 
4.13 
4.12 
4.12 
4.12 
4.J2 
4.11 
4.11 

.919816 
.949752 
.949688 
.949623 

.949494 
.949429 
.949364 

.if/ 
.07 
.07 
.08 
.08 
.03 
.08 
.08 

.707478 
707790 
.7081  02 
.708414 
.708726 
.709037 
.709349 
.709660 

5>20 
5.20 
5.20 
6.20 
6.19 
6.19 
5.19 

.292522 
.292210 
.291893 
.291586 
.291274 
.290963 
.290651 
.290340 

69 
58 
57 
66 
65 
64 
63 
52 

9 

.659271 

4.11 
4.10 

.949300 

.08 
.08 

709971 

5.19 
6.18 

.290029 

51 

10 
11 
12 
13 
14 
16 
16 

9659517 
.659763 
.660009 
.660255 
.660501 
.660746 
.660991 

4.10 
4.10 
4.10 
4.09 
4.09 
4.09 

A  AQ 

9.949235 
.949170 
.949105 
.949040 
.948975 
.948910 
.948345 

1.08 
1.08 
1.08 
1.08 
1.08 
1.08 

9.710282 
.710593 
.710904 
.711215 
.711525 
.711836 
.712146 

6.18 
6.18 
6.18 
6.18 
5.17 
6.17 

61  T 

0.289718 

.289407 
.289096 

.288785 
.288475 
.288164 
.287854 

60 
49 
48 
47 
46 
45 
44 

17 
18 

.661236 
.661481 

3.UCJ 

4.03 

A  it~ 

.948780 
.948715 

.09 
1.09 
Ino 

.712456 
.712766 

.  17 

6.17 

617 

.287544 
.287234 

43 
42 

19 

.661726 

£.Uo 

4.08 

.948650 

.uy 
1.09 

.713076 

.  1  / 

6.16 

.286924 

41 

20 

9.661970 

4  O7 

9.948584 

9.713386 

fi  Iti 

0.286614 

40 

21 
22 
23 
24 

.662214 
.6624;">9 
.662703 
.662946 

£.U/ 

4.07 
4.07 
4.06 

.948519 
.943454 
.943338 
.943323 

L09 
.09 
.09 

.713696 
.714005 
.714314 
.714624 

O.  IO 

6.16 
6.16 
6.15 

61  e 

.2863(4 
.285995 
.2856,36 
.285376 

39 
38 
37 
36 

25 
26 
27 
23 
29 

.663190 
.663433 
.66:«77 
.663920 
.664163 

4.06 
406 
4.05 
4.05 
4.05 
4.05 

.943257 
.948192 
.943126 
.943060 
.947996 

.09 
.09 
.09 
.09 
.09 
.10 

.714933 
.715242 
.715551 
.715860 
.716168 

.  15 

6.15 
6.15 
6.15 
(.14 
6.14 

.285067 
.284758 
.284449 
.284140 
.283832 

36 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.664406 
.664643 
.664391 
.635133 
.665375 
.665617 
.665359 
.666100 
.666*42 
.666533 

4.04 
4.04 
4.04 
4.03 
4.03 
4.03 
4.03 
4.02 
4.02 
4.02 

9.947929 
.947363 
.947797 
.947731 
.917665 
.947600 
.947533 
.947467 
.947401 
.947335 

.10 
.10 
.10 
.10 
.10 
.10 
.10 
.10 
.10 
.10 

9.716477 
.716785 
.717093 
.717401 
.717709 
.718017 
.713325 
.718633 
.718940 
.719243 

6.14 
6.14 
6.14 
6.13 
5.13 
5.13 
5.13 
5.13 
5.12 
5.12 

0.283523 
.283215 

.282907 
.282599 
.282291 
.281983 
.281675 
.281367 
.281060 
.280752 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 

i  43 

9.666824 
.667065 
.667305 
.667546 

4.01 
4.01 
4.01 

9.947269 
.947203 
.947136 
.947070 

.10 

.11 
.11 

9.719555 
.719362 
.720169 
720476 

5.12 
6.12 
6.11 

0.280445 
.280138 
.279331 
.279524 

20  ! 
19 
13 
17 

44 

45 
46 

.667786 
.663027 
.663267 

4.01 
4.00 
4.00 

A  Of\ 

.947004 
.946937 
.946871 

.11 

.11 
.11 

.720783 
.721089 
.721396 

5.11 
6.11 
5.11 

C  1  I 

.279217 
.278911 

.278604 

16 
15 
14 

47 

43 
49 

.668506 
.663746 
.668936 

4.UU 

3.99 
3.99 
3.99 

.946304 
^946671 

, 

.11 
.11 
.11 

.721702 
.722009 
.722315 

O.I  1 

5.10 
6.10 
6.10 

.278298 
.277991 
.277685 

13 
12 
11 

50 
51 
52 
53 
64 
65 

9.669225 
.669464 
669703 
.669942 
.670181 
.670419 

3.99 
3.98 
3.93 
3.98 
3.98 

9.946604 
.946533 
.946471 
.946404 
.946337 
.946-270 

.11 
.11 
.11 
.11 

.12 

9.722621 
.72*87 
.723232 
.723533 

.72;i344 
.724149 

6.10 
6.10 
6.09 
6.09 
5.09 

0.277379 
.277073 
.276768 
.276462 
.276156 
.275351 

10 
9 
8 
7 
6 
6 

56 
57 

.670658 
.670396 

3^97 

O  QT 

.946203 
.946136 

.12 
.12 

.724454 
.724760 

5.09 
5.09 

.275.546 
.275240 

4 
3 

68 
5S 
60 

.671134 
671372 
.671609 

O.97 
3.96 
3.96 

.946069 
.946002 
.945935 

.12 
.12 
12 

.725065 
.725370 
.72r674 

5.08 
5.08 
5.03 

.274935 
.274630 
.274326 

2 
0 

M 

Cosine. 

D.I". 

Sine. 

D  1". 

Cotaug. 

D.I'. 

Tang. 

M. 

LOGARITHMIC 


M. 

Sine. 

D.  1". 

Codne. 

D.I" 

Tang. 

D.  I". 

Cotang. 

M. 

I 

9.671609 
.671847 
.672084 
.672321 

3.96 
3.96 
3.95 

9.945935 
.945868 
.945800 
.945733 

1.12 
1.12 
1.12 

9.725674 
.725979 
.726284 

.726588 

6.08 
5.08 
6.07 

0.274326 
.274021 
.273716 
.273412 

59 
58 
57 

4 
5 

6 

7 

9 

.672558 
.672795 
.673032 
.673268 
.673505 
.673741 

3  95 
395 
394 
394 
3.94 
3.94 
3.93 

.945666 
.945598 
.945531 
.945464 
.945396 
.945328 

1.12 
1.12 
1.12 
1.12 
1.13 
1.13 
1.13 

.726892 
.727197 
.727501 
.727805 
.728109 
.728412 

5.07 
6.07 
6.07 
5.07 
6.06 
6.06 
6.06 

.273108 
.272803 
.272499 
.272195 
.271891 
.271588 

56 
CC 
54 

53 
52 
51 

10 

9.673977 

9.945261 

11  O 

9.728716 

£  f\M 

0.271284 

60 

11 
19 

.674213 
.674448 

393 

.945193 
.945125 

.Id 

1.13 

11  'J 

.729020 
.729323 

5.06 
6.06 

K.  ne 

.270980 
.270677 

49 

48 

13 
14 

.674684 
.674919 

3.92 

.945058 
.944990 

.19 

1.13 

.729626 
.729929 

o.Uo 
6.05 

.270374 
.270071 

47 
46 

1C 
16 
17 
IS 
19 

.675155 
.675390 
.675624 
.675859 
.676094 

3.92 
3.92 
3.91 
3.91 
391 
3.91 

.944922 
.944854 
.944786 
.944718 
.944650 

i'i3 

1.13 
1.13 
1.13 
1.13 

.730233 
.730535 
.730838 
.731141 
.731444 

5.05 
6.05 
6.05 
6.05 
6.04 
C.04 

.269767 
.269465 
.269162 
.268859 
.268556 

46 
44 
43 
42 
41 

90 
91 
99 
93 
94 
96 

9.676328 
.676562 
.676796 
.677030 
.677264 
.677498 

3.90 
3.90 
3.90 
390 

3.89 

o  on 

9.944582 
.944514 
.944446 
.944377 
.944309 
.944241 

1.14 
1.14 
1.14 
1.14 
1.14 

9.731746 
.732(V48 
.732351 
.732653 
.732955 
.733257 

6.04 
6.04 
6.04 
6.03 
6.03 

K  no 

0.268254 
.267952 
.267649 
.267347 
.267045 
.266743 

40 

37 
36 
36 

96 

97 
98 

.677731 
.677964 
.678197 

o  8y 
389 
3.83 

.944172 
.944104 
.944036 

1.14 
1.14 
1.14 

.733558 
.733860 
.734162 

O.Uo 
6.03 
6.03 

.266442 
.266140 
.265838 

34 

33 
39 

99 

.678430 

3-88 
3.88 

.943967 

1.14 
1.14 

.734463 

6.02 
6.09 

.965537 

31 

30 

9.67S663 

9.943899 

9.734764 

K.  no 

0.265236 

30 

31 

.678895 

3.88 

.943830 

1.14 

.735066 

Q.U* 

.264934 

29 

39 
33 
34 
36 
36 
37 
38 
39 

.679128 
.679360 
.679592 
.679824 
.680056 
.680288 
.680519 
.680750 

3.87 
3.87 
3.87 
3.87 
3.86 
386 
386 
386 
3.85 

.943761 
.943693 
.943624 
.943555 
.943486 
.943417 
.943348 
.943279 

1.14 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.16 

.735367 
.735668 
.735969 
.736269 
.736570 
.736870 
.737171 
.737471 

6.02 
6.02 
6.01 
6.01 
6.01 
6.01 
6.01 
6.01 
6.00 

.264633 
.264332 
.264031 
.263731 
.263430 
.263130 
.262829 
.262529 

28 
27 
26 
26 
24 
23 
22 
21 

40 

9.680982 

9.943210 

9.737771 

0.262229 

90 

41 
42 
43 
44 
45 
46 
47 
18 
49 

.681213 
.681443 
.681674 
.681905 
682135 
.682365 
.682595 
682825 
.683055 

3.85 
3.85 
3.84 
3.84 
3.84 
3.84 
3.83 
3.83 
3.83 

.943141 
.943072 
.943003 
.942934 
.942864 
.942795 
.942726 
.942656 
.942587 

1.15 
1.15 
1.15 
1.15 
1.15 
1.16 
1.16 
1.16 
1.16 

.738071 
.738371 
.738671 
.738971 
.739271 
.739570 
.739870 
.740169 
.740468 

6.00 
6.00 
6.00 
6.00 
4.99 
4.99 
4.99 
4.99 
4.99 

.261929 
.261629 
.261329 
.261029 
.260729 
.260430 
.260130 
.25983! 
.25953* 

19 
18 
17 
J6 
15 
14 
13 
12 

3.83 

1.16 

4.98 

60 
61 

9.683284 
.683514 

3.82 

9  00 

9.942517 

.942448 

1.16 

9.740767 
.741066 

4.98 

0.259233 
.258934 

V 
1 

62 

.633743 

3.  ox 

.942378 

1.16 

Iia 

.741365 

A  QQ 

.258635   8 

63 
64 

65 
66 
67 
68 
59 
60 

.683972 
.684201 
.684430 

.'684887 
.685115 
.685343 
.685571 

a82 
3.81 
3.81 
3.31 

3.80 
3.80 
3.80 

.942308 
.942239 
.942169 
.942099 
.949029 
.941959 
.941889 
.941819 

.18 
1.16 
1.16 
1.16 
1.16 
1.17 
1.17 
1.17 

.741664 
.741962 
.742261 
.742559 
.742858 
.743156 
.743454 
.743752 

4.99 

4.98 
4.98 
497 
497 
497 
497 
4.97 

.258336 
.258038 
.257739 
.257441 
.257142 
.256S44 
.256546 
.256248 

7 
6 
6 
4 
3 
2 
1 
0 

K. 

Oodue. 

D  1". 

Slue. 

D.  1". 

Cotang. 

D.I". 

Tang. 

M. 

61° 


COSINES,    TANGENTS,    AND    COTANGENTS. 


M 

State. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

2 
3 
4 
5 
I 
7 
8 
9 

9.685571 
635799 
.686027 
686254 
.6-86482 
.6S6709 
.686936 
687163 
.637339 
.637616 

3.80 
3.79 
3.79 
3.79 
3.79 
3.73 
3.73 
3.78 
3.73 
3.77 

9.941819 
.941749 
.941679 
.941609 
.941539 
.941469 
.941398 
.941328 
.941258 
.941137 

1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 

9.743752 
.744050 
.744343 
.744645 
.744943 
.745240 
.745538 
.745835 
.746132 
.746429 

4.96 
4.96 
4.96 
4.96 
496 
4.96 
4.95 
4.95 
4.95 
4.95 

0.256248 
.255950 
.255652 
.255355 
.255057 
.254760 
.254462 
.254165 
.253868 
.253571 

60 
69 
68 
67 
66 
66 
64 
53 
52 
61 

IG 
il 
12 

9.637843 
.  68-3069 
.633295 

3.77 
3.77 

9.941117 
.941046 
.910975 

1.18 
1.18 

9.716726 
.747023 
.747319 

4.95 
4.95 

0.253274 
.252977 
.252681 

60 
49 

48 

13 
14 
15 
16 
17 
18 
19 

.638521 
.688747 
.633972 
.689193 
.639423 
.689648 
.639373 

3.77 
3.76 
3.76 
3.76 
3.76 
3.75 
3.75 
3.75 

.940905 
.9408.34 
.940763 
.940693 
.940622 
.940551 
.940480 

1.18 
1.18 
1.18 
1.13 
1.18 
1.18 
1.18 

.747616 
.747913 
.748209 
.748505 
.748801 
.749097 
.749393 

4.94 
4.94 
4.94 
494 
4.93 
4.93 
4.93 

.252334 

.252087 
.251791 
.251495 
.251199 
.250903 
.250607 

47 
46 
45 
44 
43 
42 
41 

20 
21 

i 

24 
25 
26 
27 
28 
29 

9690098 
.690323 
.690548 
.690772 
.690996 
.691220 
.691444 
.691663 
.691392 
.6921  15 

3.75 
3.74 
3.74 
374 
3.74 
373 
3.73 
3.73 
373 
3.72 

9.940409 
.940338 
.940267 
.940196 
.940125 
.940054 
.939932 
.939911 
.939-^40 
.939768 

1.18 
1.18 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 

9.749689 
.749935 
.750281 
.750576 
'  .750872 
.751167 
.751462 
.751757 
.752052 
.752347 

4.93 
4.93 
4.93 
4.92 
4.92 
492 
492 
4.92 
492 
4.91 

0.250311 
.250015 
.249719 
.249424 
.249128 
.243833 
.248538 
.24*243 
.247948 
.247653 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 

9.692339 
.692562 
.692735 
.693003 
.693231 

3.72 
3.72 
3.72 
3.71 

9.939697 
.939625 
.939554 
.939482 
.939410 

1.19 
1.19 
1.19 
1.19 

9.752642 
.752937 
.753231 
.753526 
.753820 

4.91 
4.91 
4.91 
4.91 

0.247358 
.247063 
.246769 
.246474 
.246180 

30 
29 
28 
27 

26 

35 
36 
37 
38 
39 

.693453 
.69:3676 
.693398 
.694120 
.694342 

3.71 
3.71 
3.71 
3.70 
3.70 
3.70 

.939339 
.939267 
.939195 
.939123 
.939052 

1.20 
1.20 
1.20 
1.20 
1.20 

.754115 
.754409 
.754703 
.754997 
.755291 

4.90 
4.90 
4.90 
4.90 
4.90 

.245885 
.245591 
.245297 
.245003 
.244709 

25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 

9.694.564 
.694736 
.695007 
.695229 
.695450 
.695671 

3.70 
3.69 
3.69 
3.69 
3.69 

9.933980 
.938903 
.933336 
.938763 
.938691 
.938619 

1.20 
1.20 
1.20 
1.20 
120 

1  2ft 

9.755585 
.755878 
.756172 
.756465 
.756759 
.757052 

4.89 
4.89 
4.89 

4.89 
4.89 
4  SQ 

0.244415 
.244122 
.243328 
.243535 
.243241 
.242948 

20 
19 
18 
17 
16 
16 

46 
47 

48 

.695892 
.6961  13 
.696334 

3.68 
3.68 

.933547 
.938475 
.938402 

1.20 
1.21 

.757345 
.757638 
.757931 

4.88 
4.88 

.242655 
.242362 
.242069 

14 
13 
12 

49 

.696554 

3.67 

.933330 

1.21 

.758224 

4.88 

.241776 

11 

50 
51 
52 
53 
64 
55 
56 
57 
68 
59 
60 

9.696775 
.696995 
.697215 
.697435 
.697654 
.697874 
.693094 
.693313 
.698532 
.698751 
.6981*70 

3.67 
3.67 
3.67 
3.66 
3.66 
366 
366 
3.65 
365 
3.65 

9.933258 
.938185 
.938113 

.938040 
.937967 
.937895 
.937822 
.937749 
.937676 
.937604 
.937531 

1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.22 

9.758517 
.753810 
.759102 
.759395 
.759687 
.759979 
.760272 
.760564 
.760856 
.761148 
.761439 

4.88 
4.88 
4.87 
4.87 
4.87 
4.87 
4.87 
4.87 
4.86 
4.86 

0.241483 
.241190 
.240893 
.240605 
.240313 
.240021 
.239728 
.239436 
.239144 
.238852 
.238561 

10 
9 
8 
7 
6 
6 
4 
3 
3 

0 

M. 

Ooelne.    ' 

J>.  1". 

Blue. 

D.  I". 

Coteng. 

D.  1". 

Tnng. 

M. 

1190 


260                               TABLE    II.       LOGARITHMIC    SINES, 
80°                                                                                                              149= 

M. 

Sloe. 

D.1". 

Cosine. 

D.  1". 

Tang. 

D.l«. 

Cotang. 

M 

0 

9.698970 

9.937531 

1  99 

9.761439 

0.238561 

60 

1 
3 

.699189 
.099407 
.699626 

3.64 
3.64 

.937458 
.937385 
.937312 

1  22 
1.22 

.761731 
.762023 
.762314 

4.86 
4.86 

.23->269 
.237977 
.2376M6 

69 

68 
67 

4 
6 

.699844 
.700062 
.700280 

3.64 
363 

.937238 
.937166 
.937(»92 

1.22 
1.22 

1  V9 

.7626(6 
.762897 
.763188 

4.86 

4.85 

.237394 
.237103 
236^12 

66 
55 
54 

7 
8 
9 

.700498 
.700716 
.700933 

363 
3.63 
3.62 

.937019 
.936946 
.936872 

1.22 
1.22 
1.22 

.763479 
.763770 
.764061 

485 
4.85 
4.86 

.236521 
.236230 
.2:*S'J39 

53 
62 
61 

10 
11 
12 
13 
14 
15 
16 
17 

9.77*151 
.701368 
.7015X5 
.701802 
.702019 
.702236 
.702452 
.702669 

362 
362 
362 
361 
361 
3.61 
3.61 

9.936799 
.936?25 
.936652 
.936578 
.936.105 
.936431 
.936357 
.936284 

1.22 
1.23 
1.23 
123 
1.23 
1.23 
1.23 

9.764352 
.764643 
.764U33 
.76T>224 
.765514 
.765805 
.766095 
.766385 

446 

4.84 
4.84 
4.84 
4.84 
4.84 
4.84 

0.23fi648 
.23f>.r>7 
.23.*.'  "67 
.234776 
.21*4486 
.234195 
.2XWC5 
.233615 

60 
49 
48 
47 
46 
45 
44 
43 

18 
19 

.702885 
.703101 

3.60 
3.60 

.936210 
.936136 

1.23 
1.23 

.766675 
.766965 

4.83 
4.83 
4.83 

.233325 
.233035 

42 
41 

20 
21 
22 
23 
24 
25 

9.703317 
.703533 
.703749 
.703964 
.704179 
.704395 

3.60 
359 
359 
3.59 
359 

9.936062 
.935988 
.935914 
.935840 
.935766 
.935692 

1.23 
1.23 
1  23 
123 
1.24 

9.767255 
767545 
.767834 
768124 
788414 
768703 

4.83 
483 

483 
4.82 
4*2 

0.232745 
.232455 
.232166 
231876 
.231586 
.23I2'J7 

40 
39 
38 
37 
36 
36 

26 
27 
28 
29 

.704610 
.704«25 
.705f»40 
.705254 

3.68 
358 
358 
3.58 

.935618 
.935543 
.935469 
.935395 

1.24 
1.24 
1  24 
1.24 

.768992 
.769281 
.769571 
.769860 

4  82 

ua 

482 

4.82 
4.82 

.23IU03 
.23(1719 
.23(1429 
.230140 

34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.705469 
.705683 
.705898 
.7061  12 
.706326 
.705539 
.706753 
.706967 
.707180 
.707393 

3.67 
357 
367 
3.67 
3.56 
356 
356 
3.56 
3.55 
3.65 

9.935320 
.935246 
.935171 
.935097 
.935022 
.934948 
.934873 
.934798 
.934723 
.934649 

1.24 
1.24 
1.24 
1.24 
1  24 
1.24 
1.25 
1.25 
1.25 
l.'*5 

9.770148 
.770437 
.770726 
.771015 
.771303 
.771592 
.771880 
.772168 
.772457 
.772745 

4.81 
4.81 
4.81 
481 
481 
4.81 
4.80 
4.80 
4.80 
4.80 

0.229852 
.22i<f>63 
.229274 
.228985 
.22^697 
.25M08 
.22H20 
.227832 
.227543 
.227255 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

9707606 
.707819 
.708032 
.708245 
.7&S453 
.708670 
.708882 
.70')094 
.709306 
.709518 

3.55 
3.55 
3.54 
3.54 
3.54 
3.54 
3.54 
3.53 
3.53 
3.53 

9.934574 
.934499 
.934424 
.934349 
.934274 
.934199 
.934123 
.934048 
.933973 
.933898 

1.25 
1.25 
1.25 
1.25 
1.25 
1.25 
1.25 
1.25 
1.26 
1.2B 

9773033 
.773321 
.773608 
.773896 
.774184 
.774471 
.774759 
.775046 
.775333 
.775621 

4.80 
4.80 
4.80 
4.79 
4.79 
4.79 
4.79 
4.79 
4.79 
4.78 

0.226967 
.226679 
.226392 
.226104 
.225816 
.225529 
.225241 
.224954 
.224667 
.224379 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 
52 
53 
54 

9.709730 
.709941 
.710153 
.710364 
.710575 

3.53 
352 
3.52 
3.52 

9.933822 
.933747 
.933671 
.933596 
93aS20 

1.26 
1.26 
1.26 
1.26 

9  775908 
.776195 
.7764*2 
.776768 
.777055 

4.78 

4.78 
4.78 
4.78 

0  224092 
.2238(15 
.223518 
.223232 
.222945 

10 
9 
8 
7 
6 

55 
56 

.710786 
.710997 

3.51 

933445 
933369 

1.26 

.777342 
.777628 

4.78 

.222658 
.222372 

6 
4 

57 
68 
59 
60 

.711208 
.711419 
.711629 
.711839 

3.51 
3.51 
3.61 
3.51 

933293 
933217 
.933141 
.933066 

1.26 
1.26 
1.26 

.777915 
.778201 

.778488 
.778774 

4.77 
4.77 
4.77 

.222085 
.221799 
.221512 
.221226 

3 
2 

1 
0 

M. 

Oofiine.   i 

D.I*. 

Sine. 

D.l» 

Cotang. 

D.I"    {     Tang. 

M. 

UK* 


COSINES,  TANGENTS,  AND  COTANGENTS.        261 

lio                                              148° 

M 

Bine.  |  D.I*' 

Cosine. 

D.  1". 

I*ng. 

D.  1". 

Cotang 

M, 

0 

2 
3 
4 

6 
6 
7 
8 
9 

9.711839  1 
.7121.50  ! 
.712260 
.712469 
.712679 
.712889 
.713093 
.713308 
,713517 
.713726 

3.50 
3.50 
3.50 
3.50 
3.49 
349 
349 
349 
343 
348 

9.933066 
932990 
932914 
932838 
.  932762 
932685 
932609 
.932533 
932457 
.932330 

1.27 
1.27 
1.27 
1.27 
1.27 
1.27 
1.27 
1.27 
1.27 
1.27 

9.778774 
.779060 
.779346 
.779632 
.779913 
.780203 
.780489 
.780775 
.781060 
.781346 

4.77 
4.77 
4.77 
4.76 
4.76 
4.76 
4.76 
4.76 
4.76 
4.76 

0.221226 
.220940 
.220654 
.220368 
.221  M  (32 
.219797 
.219511 
.219225 
.218940 
.218654 

60 
59 

58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 

9.713935 
.714144 
.714352 
.714561 
.714769 

348 
3.43 
3.43 
3.47 

9.932304 
.932228 
.932151 
.932075 
.931993 

1.27 
1.27 
1.28 
1.28 

9.781631 
.781916 

.782201 
.782136 
.782771 

4.75 
4.75 
4.75 
4.75 

0.218369 
218034 
.217799 
.217514 
.217229 

50 
49 

48 
47 
46 

15 
16 
17 
18 
19 

.714978 
.715136 
.715394 
.715602 
.716809 

347 
347 
346 
3.46 
3.46 

.931921 
.931345 
.931763 
.931691 
.931614 

1.28 
1.23 
1.28 
1.28 
1.28 

.783056 
.783341 
.783626 
.783910 
.784195 

4.75 
4.75 
4.74 
4.74 
4.74 

.216944 
.216659 
.216374 
.216090 
.215805 

45 
44 

43 
42 

41 

20 
21 

9.716017 
.716-224 

3.46 

9.931537 
.931460 

1.28 

9  784479 
784764 

4.74 

0.215521 
.215236 

40 
39 

22 
23 
24 

716432 
716639 
716-^6 

345 
345 

.931333 
931306 
931229 

1.28 

1.2.3 

785048 
.785332 
.785616 

4.74 

4.74 

.214952 
.214668 
.214384 

38 
37 
36 

25 

27 
28 
29 

.717053 
717-259 
.717466 
.717673 
.717879 

3.45 
345 
344 
344 
344 
3.44 

93!  152 
931075 
930993 
930921 
.930343 

1.29 
1.29 
1.29 
1.29 
1.29 

785900 
.786184 
.786463 
.786752 
.787036 

4.73 
4.73 
4.73 
4.73 
4.73 
4.73 

.214100 
213816 
.213532 
.213248 
.212964 

35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.718085 
.718291 
.718497 
.718703 
.713909 
.719114 
.719320 
.719525 
.719730 
.719935 

343 
343 
343 
3.43 
343 
342 
342 
342 
342 

9.930766 
.930638 
930611 
.930533 
.930456 
930378 
.930300 
.930223 
.930145 
.930067 

1.29 
1.29 
1.29 
1.29 
1.29 
129 
1.30 
1.30 
1.30 

9.787319 
.787603 

.78788fi 
.788170 
.788453 
.788736 
.789019 
.789302 
.789585 
.789868 

4.73 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.71 

0.212681 
.212397 
.212114 
.211830 
.211547 
211264 
.210981 
.210698 
.210415 
.210132 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.720140 
.720345 
.720549 
.720754 
.7'20958 
.721162 
.721366 
.721570 
.721774 
.721978 

3.41 
3.41 
341 
341 
3.40 
340 
340 
340 
3.39 
3.39 

9.929989 
.92991  1 
.929333 
.9-29755 
.929677 
.929599 
.929521 
.929442 
.929364 
.929286 

1.30 
1.30 
1.30 
1.30 
1.30 
1.30 
1.30 
1.31 
1.31 
1.31 

9.790151 
.790434 
.790716 
.790999 
.791-231 
.791563 
.791846 
.792123 
.792410 
.792692 

4.71 

4.71 
4.71 
4.71 
4.71 
4.70 
4.70 
4.70 
4.70 
4.70 

0.209849 
.209566 
.209284 
.209001 
.208719 
.203437 
.208154 
.207872 
.207590 
.207308 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

5C 
61 
52 
63 
54 
55 

9.722131 
.722385 
.7225,38 
.722791 
.722994 
.723197 

3.39 
3.39 
3.39 
3.38 
3.38 

9.929207 
.9-29129 
.9'29050 
.928972 
.928893 
.923815 

1.31 
1.31 
1.31 
1.31 
1.31 

9.792974 
.793256 
.793538 
.793819 
.794101 
.794383 

4.70 
4.70 
4.70 
469 
4.69 

0.207026 
.2(6744 
.206462 
.206181 

.205*99 
.205617 

10 
9 
8 
7 
6 
5 

56 

.723400 

.928738 

1.31 

.794664 

4.69 

.20533* 

4 

57 

58 

.723603 
.723305 

337 

.928657 

.923578 

1.31 

794946 
.79*227 

4.69 

.205054 
.204773 

3 
2 

59 
60 

.72401)7 
.724210 

3.37 
3.27 

923499 
.926420 

1.31 
•  1.32 

.795508 
.795789 

4.69 
4.S9 

.2W492 
.204211 

t 

M 

Oofltae. 

L.  i". 

Slue. 

D.I" 

Cotaug. 

D.  1". 

Tang 

M 

580 


TABLE 


LOGARITHMIC    SIXES, 


M. 

Bine. 

D.  1". 

Cosine. 

D.  1". 

Tang 

D.  1". 

Cotang. 

M. 

0 

9.724210 

q  qy 

9.928420 

9.795789 

A   £Q 

0.204211 

^0~ 

1 

.724412 

u.o/ 
o  07 

.9-28342 

1    "W 

.796070 

4.OO 

A   CO 

.203930 

59 

2 

.724614 
.724*16 

o.oi 
3.36 

O  Ofl 

.928263 

L32 

.?%351 
.796632 

4  Do 
469 

A    CQ 

.203649 
.203368 

58 
67 

.725017 

O.oO 
o  oc 

!928.04 

IOO 

.796913 

4.  bo 

A   CO 

.203087 

56 

.725-219 
.725420 

U.OO 

3.36 

O  Oft 

.92Si«5 
.927946 

.  O* 

1.32 

.797194 
.797474 

4.OO 

4.68 

A   £Q 

.202806 
.202526 

55 
54 

.725622 

O.OD 
Q  QC 

927^67 

,  Cj 

.797755 

4.  DO 

.202245 

63 

.725823 
.726024 

O.oO 

3.35 
3.35 

>27787 
.927708 

L32 
1.32 

.798036 
.798316 

4.68 
4.67 
4.67 

.201964 
.201684 

52 
61 

10 
11 
12 
13 
14 
16 
16 
17 
18 
19 

9726225 
.726426 
.726626 
.726827 
.727027 
.727228 
.727428 
.727628 
.727828 
.728027 

3.35 
3.34 
334 
3.34 
3.34 
3.34 
3.33 
333 
3.33 
3.33 

9.927629 
.927549 
.927470 
.927390 
.927310 
.927231 
.927151 
.927071 
.926991 
.926911 

1.32 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 

9.798596 
.798877 
.799157 
.799437 
.799717 
.799997 
.800277 
.81X1557 
.800836 
.801116 

4.67 
4.67 
4.67 
4.67 
4.67 
466 
4.66 
466 
466 
4.66 

0.201404 
.201123 
.200843 
.200563 
.200283 
.200003 
.199723 
.199443 
.199164 
.198884 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 

9.728227 

o  oo 

9.926831 

IOO 

9.801396 

0.198604 

40 

21 
22 
23 
24 
26 
26 
27 
28 
29 

.728427 
.728626 
728825 
.729024 
.729223 
.729422 
.729621 
.729820 
.730018 

OnK) 

3.32 
3.32 
332 
3.32 
331 
3.31 
3.31 
3.31 
3.31 

.926751 
.926671 
.926591 
.926511 
.926431 
.926351 
.926270 
.926190 
.926110 

.OO 

1.33 
1  33 
1.34 
1.34 
134 
1.34 
1.34 
1  34 
1.34 

.801676 
.801955 
.802234 
.802513 
.802792 
.803072 
.803351 

1803909 

4.66 
4.66 
4.66 
465 
465 
465 
4.65 
4.65 
465 
465 

.198325 
.198045 
.197766 

.197487 
.197208 
.196928 
.196649 
.196370 
.196091 

39 
38 
37 
36 
35 
34 
33 
32 
31 

80 
31 
32 
33 
34 
36 
36 
37 
88 
89 

9.730217 
.730415 
.730613 
.730811 
.731009 
.731206 
.731404 
.731602 
.731799 
.731996 

3.30 
3.30 
3.30 
3.30 
3.30 
329 
3.29 
3.29 
3.29 
3.28 

9.926029 
925949 
925S68 
.925788 
925707 
.925626 
.925545 
.925465 
.925384 
.925303 

1.34 
1.34 
1.34 
1.34 
1.35 
1.35 
1.35 
1  35 
1.35 
1.35 

9.804187 
.804466 
.804745 
.805023 
.805302 
.805580 
.805859 
.806137 
.806415 
.806693 

4:65 
4.64 
464 
4.64 
4.64 
4.64 
4.64 
464 
4.64 
4.63 

0.195813 
.195534 
.195255 
.194977 
.194698 
.194120 
.194141 
.193863 
.193586 
.193307 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 

42 

9.732193 

.732390 
.732587 

3.28 
3.28 

9.925222 
.925141 
.925060 

1.36 
1.35 

1    OK 

W  806971 
.807249 
.807527 

4.63 
4.63 

A  £O 

0.193029 
.192761 
.192473 

20 
19 
18 

43 

.732784 

O  OQ 

.924979 

l.oo 

1    OR 

807805 

4.O.J 

A   £O 

.192195 

17 

44 

46 
16 
47 

.732980 
.733177 
.733373 
.733569 

o./o 
3.27 
3.27 
3.27 

O  OT 

.924897 
.924816 
.924735 
.924654 

I.JO 

1.35 
1.35 
1  36 

.8(18083 
.808361 
.80*638 
.808916 

4.  DO 

463 
4.63 
463 

.191917 

!l913f2 
.1910H4 

16 
15 
14 
13 

48 
IS 

.733766 
.733961 

6.4.7 

3.27 
3.26 

.924572 
.924491 

1.36 
1.36 
1.36 

.809193 
.809471 

4.62 
462 
4.C2 

.190807 
.190529 

12 
11 

60 

9.734157 

Q  O  A 

9.924409 

IOC 

9.809748 

0.190252 

10 

61 
62 
63 
64 

.734353 
.734549 
.734744 
.734939 

O.  <6O 

3.26 
3.26 
3.26 

O  OK 

.924328 
.924246 
.924164 
.9'24083 

.OO 

1.36 

l  .36 

1.36 

1JC 

.610025 
.810302 
.810580 
.810857 

4.62 
4C2 
462 

.189975 
.189698 
.189420 
.189143 

9 

8 
7 
6 

66 

.7ar>l35 

O.X) 

.924001 

.JO 

.811134 

4.62 

.188866 

6 

66 

67 
68 
59 
60 

.735330 
.735525 
.735719 
.735914 
.736109 

3.25 
3.25 
3.25 
325 
3.24 

.923919 
.923*37 
.923755 
.923673 
.923591 

1.36 
1.36 
1.37 
137 
137 

.811410 
.811687 
.811964 
.812241 
.812517 

4.61 
4.61 
4fll 
461 
4.61 

.188590 
.188313 
.188036 
187759 
.187483 

4 

3 
2 
1 
0 

M. 

Cosine. 

D.  1". 

8U». 

D.  1". 

Ootaug 

D.l" 

Tang 

H 

67° 


COSINES,  TAN<;KNTS,  AND  c  OTAN<;KNTS.       263 

33°                                                 140° 

M. 

Sine.   D.  1". 

Cosine.  D.  1". 

Tang. 

D.  1". 

Cotang. 

M.  | 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

9.736109 
.736303 
.736498 
.736692 
.736886 
.737080 
.737274 
.737467 
.737661 
.737855 

324 
3.24 
3.24 
3.23 
3.23 
3.23 
3.23 
3.23 
3.22 
3.22 

9.923591 
.923509 
.923427 
.923345 
.923263 
.923181 
.923098 
.923016 
.922933 
.922851 

1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.38 

9.812517 
.812794 
.813070 
.813347 
.813623 
.813899 
.814176 
.814452 
.814728 
.815004 

4.61 
4.61 
4.61 
4.61 
4.60 
4.60 
4.60 
4.60 
4.60 
4.60 

0.187483 
.187206 
.186930 
.186653 
.186377 
.186101 
.185824 
.185548 
.185272 
.184996 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 
12 
13 
14 

9.738048 
.738241 
.738134 
.738627 

.738820 

3.22 
3.22 
3.22 
3.21 

9.922768 
.922686 
.922603 
.922520 
.922438 

.38 
.38 
.38 

.38 

9.815280 
.815555 
.815831 
.816107 
.816382 

4.60 
4.60 
4.59 
4.59 

0.184720 
.184445 
.184169 
.183893 
.183618 

50 
49 
48 
47 
46 

15 

.739013 

3.21 

.816658 

.183342 

45 

16 
17 

18 

.739206 
.739398 
.739590 

3.21 
3.21 

92227*2 
[922189 
.922106 

.38 
.38 

00 

.816933 
.817209 
.817484 

4.59 
4.59 

.18:3067 
.182791 
.182516 

44 
43 
42 

19 

.739783 

3.20 

.922023 

.38 

.817759 

4.59 

.182241 

41 

20 

9.739975 

9.921940 

9.818035 

0.181965 

40 

21 
22 
23 
24 

.740167 
.740359 
.740550 
.740742 

3.20 
3.20 
319 

.921857 
.921774 
.921691 
.921607 

.39 
.39 
.39 
1  SQ 

.818310 
.818585 
.818860 
.8191&5 

4.58 
4.58 
4.58 

4  K8 

.181690 
.181415 
.181140 
.180865 

39 
38 
37 
36 

[  25 

.740934 

.921524 

.819410 

.180590 

H5 

26 

i  27 

.741125 

.741316 

3.19 

.921441 
.921357 

1.39 

.819684 
.819959 

4.58 
4.58 

.180316 
.180041 

34 
33 

28 
29 

.741508 
.74161)9 

3.18 
3.18 

.921274 
.921190 

1.39 
1.39 

.820234 
.820508 

4.58 
4.58 

.179766 
.179492 

32 
31 

30 

9.741889 

9.921107 

9.820783 

0.179217 

30 

31 

.742080 

Q  JO 

.921023 

.821057 

.178943 

29 

:  32 

.742271 

.920939 

.821332 

.178668 

2H 

34 

j  35 

.742462 
.742652 
.742842 

3.17 
3.17 

o  17 

.920856 
.920772 
.920688 

.40 
.40 
40 

.821606 
.821880 
.822154 

4.57 
4.57 

.178394 
.178120 
.177846 

27 
26 
25 

36 
|  37 

.743033 
.743223 

3.17 

.920604 
.920520 

.40 

.822429 
.822703 

4.57 

.177571 
.177297 

24 

23 

38 
39 

.743413 
.743602 

3.16 
3.16 

.920436 
.920352 

.40 
.40 

.822977 
.823251 

4.57 
4.56 

.177023 
.176749 

22 

21 

40 
41 
42 
43 
44 
45 
46 
47 

9.743792 
.743982 
.744171 
.744361 
.744550 
.744739 
.744928 
.745117 

3.16 
3.16 
3.16 
3.15 
3.15 
3.15 
3.15 

9.920268 
.920184 
.920099 
.920015 
.919931 
.919846 
.919762 
.919677 

.40 
.40 
.40 
.41 
.41  . 
.41 
.41 

9.823524 
.823798 
.824072 
.824345 
.824619 
.824893 
.825166 
.825439 

4.56 
4.56 
4.56 
4.56 
4.56 
4.56 
456 

0.176476 
.176202 
.175928 
.175655 
.175381 
.175107 
.174834 
.174561 

20 
19 
18 
17 
16 
15 
14 
13 

1  48 
49 

.745306 
.745494 

3.14 
3.14 

.919593 
.919508 

.41 
1.41 

.825713 
.825986 

4.55 
4.55 

.174287 
»174014 

12 
11 

50 
51 
52 
53 

9.745683 
.745371 
.746060 
.746248 

3.14 
3.14 
3.14 

9.919424 
.919339 
.919254 
.919169 

.41 
1.41 
.41 

9.826259 
.826532 
.826805 

.827078 

4.55 
4.55 
4.55 

0.173741 
.173468 
.173195 
.172922 

10 
9 

8 
7 

54 

.746436   S'iS 

.919085 

.827351 

.172649 

fi 

55 
56 
57 

.746624 
.746812 
.746999 

3.13 
3.13 

.919000 
.918915 
.918830 

1.42 
1.42 

.827624 

.827897 
.828170 

4.55 
4.55 

.172376 
.172103 
.171830 

5 

4 
8 

58 
59 
60 

.747187 
.747374 
.747562 

3.12 
3.12 

.91*745 
.918659 
.91&574 

1.42 
1.42 

.828442 
.828715 
.828987 

4.54 
4.54 

.171558 
.171285 
.171013 

3 
1 

0 

M. 

Cosine. 

D.  I". 

Sine. 

D.  I". 

Cotang. 

D.  1". 

Tang. 

M. 

133= 


56' 


-!(>4                                TABLE    II         LOGARITHMIC    SiXfcS. 

340                                                                                !4ac 

M. 

Sine 

D.  J». 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

1 
2 
3 
4 
5 
6 
7 
8 
9 

9.747562 
.747749 
.747936 
.748123 
.743310 
.743497 
.748683 
.748870 
.749066 
.749243 

3.12 
3.12 
312 
311 
311 
3.11 
311 
3.11 
3.10 
3.10 

9.  9  18574 
.918489 
.918404 
.918318 
.918233 
.918147 
.9IR062 
.917976 
.917391 
.917805 

1.42 

1.42 
1.42 
1.42 
1.42 
143 
143 
1.43 
1.43 
1.43 

9.828987 
.829260 
.829532 
.829sd5 
.830077 
.830349 
.830621 
.830893 
.831165 
.831437 

4.54 
4.54 
4.54 
454 
454 
4.64 
4.53 
4.53 
4.53 
4.53 

0.171013 
.170740 

.17H468 
.170196 
.169923 
.169651 
.169379 
.169107 
.163835 
.163563 

60 
59 
68 
67 
66 
66 
64 
53 
62 
61 

10 
11 
12 
13 
14 
16 
16 

9.749429 
.749615 
.749801 
.749987 
.750172 
.750358 
.750543 

3.10 
3.10 
3.10 
3.10 

3.09 
3.09 

9.917719 
.917634 
.917548 
.917462 
.917376 
.917290 
.917204 

1.43 
1.43 
1.4:5 
1.43 
1.43 
1.43 

9.831709 
.831981 
.832253 
.832."25 
.832796 
.833063 
.833339 

4.53 
4.53 
4.53 
4.53 
4.63 
4.53 

0.168291 
.168019 
.167747 
.167476 
.167204 
.166932 
.166661 

50 
49 
48 
47 
46 
45 
44 

17 
18 
19 

.750729 
.750914 
.751099 

3.09 
3.09 
3.08 

.917118 
.917032 
.916946 

1.44 
1.44 
1.44 

.83361  1 

.833382 
.834154 

4.62 
4.52 
4.52 
4.52 

.166389 
.166118 
.165846 

43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.751231 
.7GKf>9 
.7516C4 
.751839 
.7620X3 
.752*03 
.752392 
.752576 
752760 
.752944 

3.08 
308 
3.08 
303 
307 
3.07 
3.07 
307 
307 
3.06 

9.916859 
.916773 
.916687 
.916600 
.916511 
.916427 
.91C341 
.916254 
.916167 
.916081 

1.44 
144 
1.44 
1.44 
1.44 
1.44 
1.44 
1.44 
1.45 
1.45 

9.834425 
.834696 
.834967 
.835233 
.835509 
.835780 
.836051 
.836322 
.836593 
.836864 

4.52 
4.52 
452 
4.52 
4.52 
4.52 
4.51 
4.51 
4.51 
4.51 

0.165575 
.165304 
.165033 
.164762 
.164491 
.164220 
.163949 
.163678 
.163407 
.163136 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

80 
31 
32 

9.753123 
.753312 
.753495 

3.06 

3.06 

9.915994 
.915907 
.915820 

1.45 
1.45 

9.837134 
.837405 
.837675 

4.61 
4.61 

0.162866 
.162595 
.162325 

30 
29 

28 

33 
34 
36 
36 
37 
38 
39 

.7536/9 
.753862 
.754046 
.754229 
.754412 
.764595 
.754778 

3.06 
3.05 
3.05 
3.05 
305 
3.05 
3.05 

.915733 
.915646 
.915559 
.915472 
.915aS5 
.915297 
.915210 

1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.46 

.837946 
.838216 

.838487 
.838757 
.839027 
.839297 
.839568 

4.51 
4.61 
4.61 
4.51 
4.50 
4.60 
4.50 
4.60 

.162054 
.161784 
.161513 
.161243 
.160973 
.160703 
.160432 

27 
26 
25 
24 
23 
2tt 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.754960 
.755143 
.755326 
.755508 
.755690 
.755872 
.756054 
.756236 
.756418 
.756600 

3.04 
3.04 
3.04 
3.04 
3.04 
3.03 
3.03 
3.03 
3.03 
3.03 

9.915123 
.915035 
.914948 
.914S60 
.914773 
.914685 
.914598 
.914510 
.914422 
.914334 

1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1.48 
1.46 
1  46 

9.839838 
.840108 
.840378 
.840643 
.840917 
.841187 
.841457 
.841727 
.841998 
.642266 

4.60 

4.50 
4.60 
450 
4.50 
4.49 
4.49 
4.49 
4.49 
4.49 

0.160162 

.159892 
.159622 
.159352 
.159083 
.158813 
.158543 
.158273 
.158004 
.157734 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 
61 
62 
63 
64 
66 
66 
67 
68 

9.756782 
.756963 
.757144 
.757326 
.757507 
.757688 
.757869 
.753050 
.758230 

3.02 
3.02 
3.02 
3.02 
3.02 
3.02 
3.01 
3.01 

9.914246 
.914158 
.914070 
.913982 
.913894 
.9I38GA 
.913718 
913630 
.913541 

1.47 
1  47 
1.47 
1.47 
1.47 
1.47 
1.47 
1.47 

9.842535 

.842805 
.843074 
.84a343 
.843612 
.843382 
.844151 
.844420 
.844689 

4.49 
4.49 
449 
449 
4.49 
449 
4.48 
4.48 

0.157465 
.157195 
.156926 
.156657 
.156388 
.156118 
.155849 
.155580 
.155311 

10 
9 
8 
7 
6 
5 
4 
3 
2 

69 
60 

.753411 
.758591 

3.01 

.913453 
.913365 

1.47 
1.47 

.844958 
.845227 

4.48 

.155042 
.154773 

1 
0 

M. 

Oodne. 

D.  1". 

Sine. 

D.  1". 

Cotang 

D,l». 

Tang. 

M. 

1*40 


COSINES,    TANGENTS,    AND    COTANGENTS. 


M. 

Sine. 

D.I" 

Codna. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M 

0 
1 
2 
3 
4 
6 
6 
7 
8 
9 

9.758591 
.758772 
.768952 
.759132 
.769312 
.7594'.»2 
.759672 
.759852 
.760031 
.760211 

3.01 
3.00 

3.00 
3.f»0 
3.00 
3.00 
2.99 
2.99 
2.99 
2.99 

9.913365 
.913276 
.9131*7 
.9130H9 
.913010 
.912922 
.912833 
.912744 
.912655 
.912566 

1.47 
1.48 
1  48 
1.48 
148 
1.48 
1.43 
1.48 
1.43 
1.48 

9.845227 
.845496 
.845764 
.846033 
.846302 
.846570 
.846839 
.847108 
.847376 
.847644 

4.48 
4.48 
4.48 
4.48 
4.48 
4.48 
4.48 
4.47 
447 
4.47 

0.154773 
.154504 
.154236 
.153967 
.153698 
153430 
.153161 
.152*92 
.152624 
.162356 

60 
69 

68 
67 
66 
66 
64 
63 
62 
61 

10 
11 

9.760390 
.760569 

2.99 

9.912477 
.912338 

1.48 

9.847913 

.848181 

4.47 

0.152087 
.151819 

50 
49 

12 
13 

14 
15 
16 
17 
18 
19 

.760748 
.760927 
.761106 
.761285 
.761464 
.761642 
.761821 
.761999 

2.98 
2.98 
2.98 
2.98 
2.98 
2.97 
2.97 
2.97 

.912299 
.912210 
.912121 
.912031 
.911942 
.911853 
.911763 
.911674 

1.49 
1.49 
1.49 
1.49 
1.49 
1  49 
1.49 
1.49 

.848449 
.848717 
.848986 
.649254 
.849522 
.849790 
.850057 
.350325 

447 
4.47 
4.47 
4.47 
4.47 
4.46 
4.46 
4.46 

.151651 
.151283 
.151014 
.160746 
.150478 
.160210 
.149943 
.149675 

48 
47 
46 
45 
44 
43 
42 
41 

4) 
21 
22 
23 
24 
25 
26 
17 
28 
29 

9.7C2177 
.762356 
.7625:34 
.762712 
.762889 
.763067 
.763245 
.763422 
.763600 
.763777 

2.97 
2.97 
2.97 
2.96 
296 
2.96 
2.96 
2.96 
2.95 
2.95 

9.911584 
.911495 
.911405 
.911315 
.91  1226 
.911136 
.911046 
.910956 
910866 
910776 

1.49 
1.49 
1.49 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 

9.850593 
.850861 
.851129 
851396 
.851664 
851931 
.852199 
.852466 
.852733 
.853001 

4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 

0.149407 
.149139 
.148871 
.148604 
.148336 
.148069 
.147801 
.147534 
.147267 
.146999 

40 
39 
38 
37 
36 
36 
34 
33 
32 
31 

30 
31 

9.763954 
764131 

2.95 

9.910686 
910596 

1.50 

9.853268 
.853535 

4.45 

0.146732 
.146465 

30 
29 

32 
33 
34 
35 
36 
37 
38 

764308 
764485 
.764662 
.764838 
.765015 
.765191 
.765367 

2.95 
295 
2.94 
2.94 
2.94 
2.94 

910506 
910416 
.910325 
.910235 
910144 
.910054 
.909963 

1.60 
1.61 
1.61 
1.61 
1.61 
1.61 

.853802 
.854(69 
.854336 
.854603 
.854870 
.855137 
.855404 

4.45 
4.45 
4.45 
4.45 
4.45 
4.45 

.146198 
.145931 
.145664 
.146397 
.145130 
.144863 
.144596 

28 
27 
26 
26 
24 
23 
22 

39 

.765544 

2.03 

.909873 

1.51 

.856671 

4.44 

.144329 

21 

40 
41 
42 

9.765720 
.765896 
.766072 

2.93 
2.93 

9.909782 
.909691 
.909601 

1.51 
1.C1 

9.855938 
.856204 
.856471 

4.44 

4.44 

0.144062 
.143796 
.143529 

20 
19 

18 

43 

.766247 

.909510 

1.51 

.856737 

.143263 

17 

44 

45 
46 
47 
48 
49 

.766423 
.766593 
.766774 
.766949 
.767124 
.767300 

2.93 
292 
2.92 
2.92 
2.92 
2.92 

909419 
909328 
.909237 
.909146 
.909055 
.9089^4 

1  51 
1  52 
1.6& 
1.62 
1.62 
1.62 
1.52 

.857004 
.857270 
.857537 
.867803 
.858069 
.858336 

4.44 
4.44 
4.44 
4.44 
444 
4.44 

.142996 
.142730 
.142463 
.142197 
.141931 
.141664 

16 
15 
14 
13 
12 
11 

60 
61 
62 
53 
64 
65 
56 
67 
63 
59 
60 

9767475 
.767649 
.767824 
.767999 
.768173 
.768348 
.76>522 
.76*897 
.768871 
.769045 
.769219 

2.91 
291 
291 
2.91 
291 
2.91 
2.90 
290 
2.90 
2.90 

9.908873 
.90S781 
.908690 

.'90*507 
.90S4I6 
.90S324 
.908233 
.90S141 
.908049 
.907958 

1.62 
1.52 
1.62 
1.62 
1.52 
1.53 
1.53 
1.53 
1.53 
1.53 

9.858602 
.858868 
.859134 
.859400 
.859666 

!860198 
.860464 
.860730 
.860995 
.861261 

4.44 
443 
443 
443 
4.43 
443 
443 
4.43 
4.43 
4.43 

0.141398 
.141132 
.140866 
.140600 
.140334 
.140068 
.139802 
.139536 
.139270 
.139005 
.138739 

10 
9 
8 
7 

0 

M. 

OOBUM. 

D.1". 

Blue. 

D.I". 

CoMng. 

D.  1". 

Tang; 

M. 

5*4 


266 


TABLE    II.       LOGARITHMIC    SINES, 


M. 

Sloe. 

D.  1". 

Ooeine. 

D.  1". 

Tkmg. 

D.  1". 

Ootang. 

M 

0 

9.769219 
.769393 
.769566 
.769740 

2.90 
2.90 
2.89 

9.907958 
.907866 
.907774 
.907682 

1.53 
1.53 
1.53 

9.861261 
.861527 
.861792 
.862058 

4.43 
4.43 
4.43 

0.138739 
.138473 
.138208 
.137942 

60 
59 

58 
57 

.769913 

.907590 

.862323 

.137677 

56 

.770087 
.770260 
.770433 
.7706(* 
.770779 

2.89 
2.89 

2.88 
2.88 
2.88 

.907498 
.907406 
.907314 
.907222 
.907129 

1.53 
1.54 
1.54 
1.54 
1.54 

.862589 
.862854 
.863119 
.863385 
.863650 

4.42 
4.42 
4.42 
4.42 
4.42 

.137411 
.137146 
.136881 
.136615 
.136350 

55 
54 
53 
Si 
51 

10 

11 

9.770952 
.771125 

2.88 

9.907037 
.906945 

1.54 

9.863915 
.864180 

4.42 

0.136085 
.135820 

50 

49 

12 
13 
14 
15 
18 
17 
18 
19 

.771298 
.771470 
.771643 
.771815 
.771987 
.772159 
.772331 
.772503 

2.88 
2.87 
2.87 
2.87 
2.87 
2.87 
2.87 
2.86 

.906852 
.906760 
.906667 
.906575 
.906482 
.906389 
.906296 
.906204 

1.54 
1.54 
1.54 
1.54 
1.55 
1.55 
1.55 
1.55 

.864445 
.864710 
.864975 
.865240 
.865505 
.865770 
.866035 
.866300 

4.42 
4.42 
4.42 
4.41 
4.41 
4.41 
4.41 
4.41 

.135555 
.135290 
.135025 
.134760 
.134495 
.134230 
.133965 
.133700 

48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.772675 
.772847 
.773018 
.773190 
.773361 
.773533 
.773704 
.773375 
.774046 
.774217 

2.86 
2.86 
2.86 
2.86 
2.85 
2.85 
2.85 
2.85 
2.85 
2.85 

9.906111 
.906018 
.905925 
.905832 
.905739 
.905645 
.905552 
.905459 
.905366 
.905272 

1.55 
1.55 
1.55 
1.55 
1.55 
1.55 
155 
1.56 
1.56 
1.56 

9.866564 
.866829 
.867094 
.867358 
.867623 
.867887 
.868152 
.868416 
.868680 
.868945 

4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.40 
4.40 

0.133430 
.133171 
.132906 
.132642 
.132377 
.132113 
.131848 
.131584 
.131320 
.131055 

40 
39 
38 

37 
36 
35 
34 
33 
32 
31 

30 
31 
32 
33 
34 

9.774388 
.774558 
.774729 
.774899 
.775070 

2.84 
2.84 
2.84 
2.84 

9.905179 

•905085 
.904992 
.904898 
.904804 

1.56 
.56 
.56 
.56 

9.869209 
.869473 
.869737 
.870001 
.870265 

4.40 
4.40 
4.40 
4.40 

0.130791 
.130527 
.130263 
.129999 
.129735 

30 
29 
28 
27 
26 

35 
36 
37 

38 
39 

775240 
.775410 
.775680 
.775750 
.775920 

2.84 
2.83 
2.83 
2.83 

.904711 
.904617 
.904523 
.904429 
.904335 

.56 
.56 

.57 
.57 

.870529 
.870793 
.871057 
.871321 
.871585 

4.40 
4.40 
4.40 
4.40 

.129471 
.129207 
.128943 
.128679 
.128415 

25  1 
24 
23 
22 
21 

40 
41 
42 
43 
44 

9.776090 
.776259 
.776429 
.776598 
.776768 

283 
2.83 
2.82 

2.82 

9.904241 
.9<H147 
.904053 
.903959 
.903864 

.67 
.67 
.57 
.57 

9.871849 
.872112 
.872376 
.872640 
.872903 

4.40 
4.39 
4.39 
4.39 

0.128151 

.127888 
.127624 
.127360 
.127097 

20 
19 

18 
17 
16 

45 
46 
47 

48 
49 

.775937 
.777106 
.777275 

.777444 
.777613 

2.82 
2.82 
2.82 
2.81 
2.81 

.903770 
.903676 
.903581 
.903487 
.903392 

57 
.57 
.57 
.67 

.58 
.58 

.873167 
.873430 
.873694 
.873957 
.874220 

4.39 
4.39 
4.39 
4.39 
439 
4.39 

.126833 
.126570 
.126306 
.126043 
.12578U 

15 
14 
13 
12 
11 

50 
51 
52 

9.777781 
.777950 
.778119 

2.81 
2.81 

9.903298 
.903203 
.903108 

.58 
.58 

9.874484 
.874747 
.875010 

4.39 
439 

0.125510 
.125253 

.1249*1 

10 
9 

8 

53 

.778287 

.903014 

.875273 

.124727 

7 

54 

55 
56 

.778455 
.778624 
.778792 

2.80 
2.80 

.902919 
.902824 
902729 

.58 

.58 

.875537 
.875800 
.876063 

4.39 
4.38 
433 

.124163 
.124200 
.123937 

6 

5 

4 

57 
68 

.778960 
.779128 

2.80 

.902334 
.902539 

.58 

.876326 
.876589 

4.38 
4.38 

.123674 
.123411 

8 
1 

59 
60 

.779295 
.779463 

2.79 

.902444 
.902349 

.59 

.876852 
.877114 

438 

.123148 

.122886 

I 
0 

M. 

Oodnn. 

D.  1". 

Blue. 

D.  1". 

Ootaug. 

D.  1". 

Tang. 

M. 

53P 


COSINES,    TANGENTS,    AND    COTANGENTS. 


267 


M. 

Bine. 

D.l*. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

If. 

0 

9.779463 

9  70 

9.9023-19 

ICQ 

9.877114 

*438 

0.122886 

60 

1 

.779631 

z.  /y 

o  70 

.902*53 

.Oil 

Irn 

.877377 

3..X5 
A  *}Q 

.122623 

69 

2 
3 

.779798 
.779966 

K./9 

2.79 

2-*O 

.902158 

.Oi7 

1.59 

ICQ 

.877640 
.877903 

4.  Jo 
4.33 

.122360 
.122097 

58 
57 

4 
6 
6 

.780133 
.780300 
.780487 

'  *  J 

279 

2.78 

!901967 
.901872 
.901776 

&y 
1  59 
159 

.878165 
.876428 
.87S691 

4.38 
4.38 
438 

.121835 
.121572 
.121309 

56 
55 
64 

7 

8 
9 

.780634 
.781  Mil 
.780968 

2^78 
278 
2.78 

.901631 
.901585 
.901490 

1.59 
1.59 
1.60 

.873953 
.879216 

.879478 

*38 
4.37 
4.37 

.121047 
.120784 
.120522 

53 
62 
61 

10 
11 
12 

9.781134 
.781301 
.781468 

2.78 
2.77 

2""V 

9.901394 
.901  '293 
.901202 

1.60 
160 

9.879741 

.880003 
.880265 

4.37 
4.37 

0.120259 
.119997 
.119735 

50 
49 

48 

13 

.78*634 

.47 

o  77 

.901106 

1  60 

.880528 

4.37 

.119472 

47 

14 

.781800 

X.  /  / 

2~"7 

.901010 

I*?? 

.880790 

4  37 

.119210 

46 

15 

.781966 

.  t  t 
2Mta 

.900914 

1.60 

.881052 

4.37 

.118948 

45 

16 

.782132 

.// 

277 

.900813 

1.60 

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4.37 

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44 

17 

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.834577 

4.37 

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43 

18 

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.900626 

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.881839- 

4.37 

.118161 

42 

19 

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2^76 

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1.61 

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4.37 
4.37 

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41 

20 

9.782796 

2  76 

9.900433 

1  £1 

9.882363 

0.117637 

40 

21 

.782961 

27fi 

.900337 

1  ol 

1c  i 

.832625 

4.37 

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39 

22 

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.  /o 

O  7*5 

.900240 

ol 

1c.\ 

.882837 

4.37 

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38 

23 
24 

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.783458 

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2.75 

27c 

.9001  M 
.900047 

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1  61 

1C  1 

.883148 
.883410 

4.36 
4.36 

•  .116852 
.116590 

37 
36 

25 

.783623 

.  <  O 
O  7K 

.899951 

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.883672 

4.36 

,116328 

35 

26 

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.899854 

1.61 

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.a33934 

4.36 

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34 

27 

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.  fO 

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.899767 

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33 

28 
29 

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2.74 

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1.61 
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4.36 
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32 
31 

30 
31 

9.784447 
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274 

274 

9.899467 
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1.62 

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0.115020 
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30 
29 

32 

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.  t\ 

2*fA 

.899273 

I  O4 

.885504 

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.114496 

28 

33 

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1.62 

.885765 

4.36 

.114235 

27 

34 

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2.74 

.899078 

1  62 

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4.36 

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20 

35 

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2.74 

O  "^O 

.893981 

1  .62 

ICO 

.886288 

4.36 

.113712 

25 

36 
37 

.785433 
.785597 

».  To 

2.73 

O  TO 

.893384 
.898787 

.b.s 
1  62 

.886549 
.886811 

4.36 
4.36 

.113451 
.113189 

24 
23 

38 
39 

.785761 

.785925 

£>.  t  O 

2.73 
2.73 

.893639 
.898592 

l'62 
1.62 

.887072 
.887333 

4.35 
4.35 
4.35 

.112928 
.112667 

22 
21 

40 
41 
42 
43 
44 
45 

9  786039 
.786252 
.786416 
.786579 
.786742 
.786906 

2.73 

2.73 
272 
272 
272 

Q  7O 

9.898494 
.893397 
.898299 
.898202 
.893104 
.898006 

1.63 
1  63 
1.63 
1.63 
1.63 

9.887594 

.887855 
.888116 
.888378 
.888639 
.888900 

4.35 
4.35 
4.35 
4.35 
4.35 

0.112406 
.112145 
.111884 
.111622 
.111361 
.111100 

20 
19 
18 
17 
16 
15 

43 
47 
48 
49 

.787069 
.737232 
.787395 
.787557 

•c.  t  ft 

2.72 
2.72 
2.71 
2.71 

.897908 
.897810 
.897712 
.897614 

1.63 
1.63 
1  63 
1.63 

.889161 
.889421 
.889632 
.889943 

4.35 
4.35 
4.35 
4.35 
4.35 

.110839 
.110579 
.110318 
.110057 

14 
13 
12 
11 

60 
61 

9.787720 

.787883 

271 

9.897516 
.897418 

1.64 

9.890204 
.890465 

4.35 

O.f09796 
.109535 

10 
9 

62 
53 
54 
55 
66 
57 
68 
59 
60 

.788045 
.788208 
.788370 
.788532 
783694 
.783856 
789018 
.789180 
.789342 

2  71 
271 
271 
270 
270 
2.70 
270 
2.70 
2.70 

.897320 
.897222 
.897123 
.897025 
.896926 
.896>*iS 
.896729 
.896631 
.896532 

1  64 
1  64 
1.64 
1  64 
1  64 
1.64 
1.64 
1  64 
1.64 

.890725 
.890936 
891247 
.891507 
891763 
892028 
.892289 
.892549 
.892810 

4.35 
434 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 

.109275 
.109014 
.108753 
.10*193 
.103232 
107972 
.107711 
.107451 
.107190 

8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Oodno. 

D.I". 

Sine. 

D.  1". 

Cotaug 

D.  1". 

Tang. 

M. 

1587' 


268 


TAJiLE    II.       LOGARITHMIC    SINES. 


1410 


M. 

Blue. 

D.  1". 

Conine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 
1 
2 
3 
4 
5 
8 
7 
8 
9 

9.789342 
.7895(>4 
.789665 
.789827 
.789988 
.790149 
.790310 
.790471 
.790632 
.790793 

2.69* 
2.69 
269 
269 
269 
2.69 
268 
268 
263 
2.68 

9.696532 
.896433 
.896335 
.896236 
.896137 
.8960:38 
.895939 
.895*40 
.895741 
.895641 

1.65 
1.65 
1.65 
165 
.65 
.65 
.65 
.65 
65 
.65 

9.892*10 
.893070 
.693331 
.893591 

!6941I1 
.694372 
.894632 
.894392 
.895152 

434 
4.34 
434 
434 
434 
434 
434 
434 
4.33 
4.33 

0.107190 
.106930 
.106669 
106409 
.106149 
.105*89 
.10»628 
.105368 
.105108 
.104848 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 
11 

9790954 
.791115 

268 

9.895542 
.895443 

.66 

9.895412 
.895672 

4.33 

0.104588 
.104328 

60 
49 

12 
13 
14 
15 
16 
17 
18 
19 

.791275 
.791436 
.791596 
.791757 
.791917 
.792077 
.792237 
.792397 

267 
267 
267 
267 
2.67 
267 
267 
2.66 

.895343 
.895244 
.895145 
.895(H5 
.894945 
.894S46 
.894716 
.894646 

.66 
.66 
.66 
.66 
.66 
66 
66 
.66 

.895932 
.896192 
.896452 
.896712 
.696971 
.897231 
.897491 
.897751 

433 
433 
433 
433 
433 
433 
433 
4.33 

.1041168 
.103808 
.103548 
.1032*8 
.103029 
.102769 
.  102509 
.102249 

48 
47 
46 
45 
44 
43 
42 
41 

20 

22 
23 
24 
26 
26 
27 
28 
29 

9.792557 
.792716 
.792876 
.793035 
.793195 
.793354 
.793514 
.793673 
.793832 
.793991 

266 
266 
2H6 
266 
266 
265 
265 
265 
265 
2.65 

9.89-1546 
.8'J4!  46 
.894346 
.894246 
.894146 
.694046 
.893946 
.893846 
.893745 
.893645 

.67 
67 
.67 
.67 
.67 
67 
67 
.67 
.67 
.67 

9.898010 
.898270 
.898530 
.898789 
.899049 
.891)303 
.699563 
.899827 
.9000H7 
.900346 

433 
433 
433 
433 
433 
4  32 
4  32 
43* 
4  32 
432 

0.101990 
.101730 
.101470 
.101211 

.looar.i 

.10061)2 
.100432 
.100173 
.099913 
.099654 

40 
39 
38 
37 
36 

a 

33 
32 
31 

30 
31 
82 
33 
34 
35 
36 
37 
38 
39 

9794150 
.794308 
.794467 
.794626 
.794784 
.794942 
.795101 
.795259 
.795417 
.795575 

265 
2.64 
2.6-1 
264 
264 
264 
264 
264 
2f,3 
2.63 

9.893544 
.693414 
.693343 
.893243 
.693142 
.893041 
.892U40 
.892*39 
.892739 
.892638 

.68 
.63 
.68 
.63 
68 
.68 
.63 
68 
66 
.68 

9.900605 
.900SG4 
.901124 
.9013*3 
.901642 
.901901 
.902160 
.902120 
.902679 
.902933 

432 
432 
4  32 
432 
432 
432 
432 
432 
432 
4.32 

0.099395 

!09**76 

.09*617 
.098358 
.09*099 
.01)7^0 

!09732I 
.097062 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

40 
41 

9.795733 
.795891 

263 

9.892536 
.692435 

69 

9.903197 
.903456 

432 

ioyr.544 

20 
19 

42 
43 

.796049 
.796206 

263 

.6923:34 
.892233 

.69 

.903714 
903973 

4  31 

18 
17 

44 
45 
46 

.796364 
.796521 
.796679 

262 
2.62 

.892132 
.692030 
.891929 

.69 
.69 

.904232 
.904491 
.904750 

431 
431 

.095768 
.095509 

16 
15 
14 

47 

48 

.796336 
.796993 

262 

^891726 

69 

.905003 
.905267 

4  31 

>>94733 

IB 
12 

49 

.797150 

2.61 

.891624 

.69 

.905526 

4.31 

.094474 

i. 

60 
51 
52 
53 

9.797307 
.797464 
.797621 

.797777 

2.61 
261 
261 

9.89IT.23 
.891421 
.891319 
.891217 

.70 
.70 
.70 

9.9057«5 
.906043 
.«X«3i  >2 

.9f'G56(» 

431 
431 
4.31 

0.094215 
.093957 
.093698 
.093440 

10 
9 

8 
7 

54 
55 

.797934 
.79^091 

261 

.891115 
.891013 

70 

.906s  1  9 
.907077 

431 

.093  HI 
.0921)23 

6 
5 

1  56 
57 
68 

.79^247 
.79«403 
.798560 

261. 
260 

.890911 
.890^09 
.890707 

.70 
.70 

.907336 
.907594 
.907853 

4.31 
431 
431 

.0926B4 
.01)2406 
.092147 

4 

3 
2 

59 

.793716 

~  5!; 

.890605 

.908111 

.091*89 

60 

.793872 

.890503 

.908369 

4.31 

.091631 

0 

M. 

(Cosine. 

D.I 

bine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

M. 

51° 


COSINES,  TANGENTS,  AND  COTANGENTS.        '269 

890                                              140° 

M. 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

I 
2 

9.798872 
.799028 
.799134 

2.60 

2.60 

9.890503 
.890400 
.890298 

1.71 
1.71 

9.90.8369 

.9US628 

4.30' 
4.30 

0.091631 
.091372 
.091114 

60 
59 
58 

3 
4 
5 
6 

7 
8 
9 

.799339 
.799495 
.799651 
.799806 
.799962 
.800117 
.800272 

2.60 
2.59 
2.59 
2.59 
2.59 
2.59 
2.59 
2.59 

.890195 
.890093 
.889990 

.8898,88 
.889785 
.889682 
.889579 

1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 

!909144 
.9UJ402 

.909660 

.910177 
.910135 
.910693 

430 
4  30 
430 
4.30  . 
4.30 
4.30 
4.30 

.090356 
.090598 
.090340 
.090082 
.089823 
.089565 
.089307 

57 
56 
55 
54 
53 
52 
51 

10 
ll 
12 
13 
14 
16 
10 
17 
18 
19 

9.800427 
.800582 
.800737 
.800892 
.W1047 
.SO  1201 
.801356 
.801511 
.801665 
801319 

2.58 
2.58 
2.58 
2.58 
2.58 
2.53 
2.57 
2.57 
2.57 
2.57 

9.889477 
.889374 

.889271 
.889168 
.889064 
.883961 
.888853 
.888755 
.888651 
.888543 

1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 

9.910951 
.911209 
.911467 
.911725 
.911982 
.912240 
.912493 
.912756 
.913014 
.913271 

4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 

0.089049 
.088791 
.088533 
.088275 
.088018 
.087760 
.087502 
.087244 
.086986 
.086729 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 

9801973 
.802123 

.802282 
.802436 
.802589 
.802743 

2.57 
2.57 
2.57 
256 
2,50 

9.888444 
.888341 

.888237 
.888134 

.888030 
.887926 

1.73 
1.73 
1.73 
1.73 
1.73 

9.913529 
.913737 
.914044 
.914302 
.914560 
.914817 

4.29 
4.29 
429 
4.29 
4.90 

0.086471 
.086213 
.085956 
.085693 
.085440 
.085183 

40 
39 

38 
37 
36 

36 

26 
27 
23 
29 

'.803050 
.803204 
.803357 

2.56 
2.56 
2.56 
2.56 
2.55 

.887322 
.887718 
.837614 
.887510 

1.73 
1.73 
1.73 
173 
1.74 

.915075 
.915332 
.915590 
.915547 

429 
4.29 
429 
4.29 

.084925 
.0*4*68 
.084410 
.084153 

34 
33 
32 
31 

30 

98035U 

9.887406 

If  A 

9.916104 

0.083396 

30 

31 
32 
33 
34 
35 
36 

.803664 
.803-117 
.803970 
.804123 
.804276 
.804428 

2.55 
2.55 
255 
2.55 
255 
2.55 

.887302 
.887193 
.887093 
.886989 
.886886 
.886780 

.74 
1.74 
1.74 
1.74 
1.74 
1.74 

.916362 
.910619 
.916877 
.917134 
.917391 
.917648 

429 
429 
4.29 
429 
4.29 

.083633 
.083381 
.083123 
.082*56 
.052609 
.082352 

29 
28 
27 
26 
25 
24 

37 
38 
39 

.8IM581 
.804734 
.804386 

2  54 
254 
2.54 
2.54 

.886676 
.886571 
.886466 

1.74 
1.74 
1.74 
1.75 

.917906 
.918163 
.918420 

4'  29 
4.29 
4.29 

.082094 
.031837 
.081580 

23 
22 
21 

40 
41 

a 

43 
44 
45 
46 

47 
48 
49 

9.805039 
.805191 
.805343 
.805495 
.805647 
.805799 
.805951 
.806103 
.806254 
.806406 

254 

254 
254 
2.53 
2.53 
2.53 
2.53 
253 
253 
2.52 

9.&S6362 

.886257 
.886152 
.886047 
.8859-42 
.8*5837 
.885732 
.8S5627 
.885522 
.885416 

1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.76 

9.918677 

.918934 
.919191 
.919443 
.919705 
.919962 
.920219 
.92IM76 
.920733 
.920990 

4.28 
423 
4.28 
4.23 
4.28 
4.28 
4.28 
4.28 
423 
4.28 

0.081323 
.081066 
.080809 
.080552 
.080295 
.0*0038 
.079781 
.079524 
.079267 
.079010 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

50 
51 

9.806557 
.800709 

2.52 

9.885311 

.8ft5205 

1.76 

9.921247 
.921503 

428 

0.078753 

.07^497 

10 
9 

52 

.806S60 

2  52 

.885100 

1.76 

.921760 

4  28 

.07-^240 

8 

53 
54 

.807011 
.807163 

2.52 
2.52 

.8*4994 

.884889 

1.76 
1.76 

.922017 
.92-2-274 

4.28 
4-23 

.0779S3 
.077726 

7 
6 

55 
56 
57 
53 
59 
80 

.807314 
.807465 
.807615 
.807766 
.807917 
808067 

2.52 
2.52 
2.51 
2.51 
251 
2.51 

.884783 
.884677 
.884572 
.884466 
.884360 
.884254 

1.76 
1.76 
1.76 
1.76 
1.77 
1.77 

.922530 
.922787 
.923044 
.923300 
.923567 
.OU3R14 

4.28 
423 
423 
428 
4.28 
4.28 

.077470 
.077213 
.076956 
.076700 
.076443 
.076186 

5 
4 
3 
2 

0 

M. 

Codno 

D.  1". 

Sloe. 

D.1". 

Cotang 

D.  1". 

Ofcng 

M. 

189° 


50° 


270 


OC.ARITH  MIC     SINES. 


M. 

Sine. 

D.  1". 

Cosine. 

D.  I". 

Tang. 

D.I  . 

Cotang. 

M. 

0 

9808067 

o  Ci 

9.884254 

9.923314 

0.076186 

60 

1 

.808213 

V.OI 

9  ci 

.884148 

77 

.924070 

A  OQ 

.075930 

59 

2 
3 
4 

.803363 
.808519 
.808669 

B.OI 

2.51 
2.50 
o  £n 

.884(142 
.883936 
.883829 

'.77 
.77 

77 

.924327 
'.924840 

V.J0 

4.27 
4.27 

A  O7 

.075673 
.075417 
.075160 

58 
57 
56 

6 
6 
7 

.808819 
803969 
.809119 

VkOU 

2.50 
250 

O  £/k 

.883723 
.883617 
.883510 

.  // 

.77 

.77 

.925(196 
.925352 
.925609 

*±.  £.1 

4.27 
4.27 

.074904 
.074648 
.074391 

55 
54 
53 

8 

.809269 

3LMJ 

.883404 

.77 

.925865 

4.27 

.074135 

52 

9 

.809419 

2.50 

.883297 

.78 
.73 

.926122 

4^27 

.073878 

51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.809569 
.809718 
.809368 
.810017 
.810167 
.810316 
.810465 
.810614 
.810763 
.810912 

2.49 
2.49 
2.49 
2.49 
2.49 
2.49 
2.43 
2.43 
2.48 
2.43 

9.883191 

.883084 
.882977 
.882371 
.832764 
.882657 
.832550 
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.78 
.78 
.78 
.78 
.78 
.78 
.78 
.79 
.79 
.79 

9.926378 
.926634 
.926390 
.927147 
.927403 
.927659 
.927915 
.923171 
.923427 
.92S684 

4.27 

4.27 
4.27 
427 
4.27 
4.27 
4.27 
4.27 
427 
4.27 

0.073622 
073366 
.073110 
.072853 
.072597 
.072341 
.072085 
.071329 
.071573 
.071316 

50 
49 
43 
47 
46 
45 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 

9.811061 
.811210 
.811358 
.811507 
.811655 
.811804 
.811952 
.812100 
.812243 

243 

2.48 
248 
2.47 
2.47 
2.47 
2.47 
247 

9882121 
.882014 
.881907 
.881799 
.881692 
.88I5H4 
.881477 
.881369 
.881261 

.79 
.79 
.79 
.79 
.79 
.79 
.79 
.80 

9.92S940 
.929196 
.929-152 
.929703 
.929964 
.930220 
.930475 
.930731 
.93U937 

427 

4.27 
4.27 
4.27 
4.27 
4.27 
4.26 
4.26 

0.071060 
.070804 
.070543 
.070292 
.070036 
.069780 
.069525 
.069269 
.069013 

40 
39 
38 
37 
36 
35 
34 
33 
32 

29 

.812396 

2.47 
2.47 

.881153 

.80 
.80 

.931243 

4  26 
4.26 

.063757 

31 

30 
31 
32 
33 

9.812544 
.812692 
.812840 
.812988 

246 

246 
2.46 

9.881046 
.880933 
.880330 
.880722 

.80 
.80 
.80 

9.931499 
.931755 
.932010 
.932266 

4:26 
4.26 
4.26 

0.068501 
.068245 
.067990 
.067734 

30 
29 
28 
27 

34 
35 

.813135 
.813283 

2.46 
2.46 

.830613 
.880505 

.80 
.80 

.932522 
.932778 

4.26 
4.26 

.067478 
.067222 

26 
25 

36 

.813430 

2.46 

O  Ifi 

.880397 

.80 

Dl 

.933033 

4.26 

.066967 

24 

37 

.813578 

2.4O 

.880289 

Jot 

.933239 

4.26 

066711 

23 

38 
39 

.813725 
.813872 

2.45 
2.45 
2.45 

.880180 
.880072 

.81 
.81 
.81 

.933545 
.933800 

4.26 
4.26 
4.26 

.066455 
.066200 

22 
21 

40 
41 

9.814019 
.814166 

2.45 

9.8^9963 

.879855 

.81 

9.934056 
.934311 

4.26 

0.065944 
.065689 

20 
19 

42 
43 
44 
45 
46 

.814313 
.814460 
.814607 
.814753 
.814900 

2.45 
2.45 
2.45 
2.44 
2.44 

.879746 
.879637 
.879529 
.870420 
.879311 

.81 
.81 
.81 
.81 
.81 

.934567 
.934822 
.935078 
.935333 
.935589 

4.26 
4.26 
4.26 
4.26 
4.26 

.065433 
.065178 
.064922 
.064667 
.064411 

18 
17 
16 
15  ' 
14 

47 

.815046 

2.44 

.879202 

.82 

.935844 

4.26 

.064156 

13 

48 

.815193 

2.44 

.879093 

.82 

.936100 

4.26 

.063900 

12 

49 

.815339 

2.44 
2.44 

.878984 

.82 
.82 

.936355 

4.26 
4.26 

.063645 

11 

50 

9.815485 

9.878875 

OO 

9.936611 

0.063389 

10 

51 
52 
53 
54 
55 

.815632 
.815778 
.815924 
.816069 
.816215 

2.44 
2.43 
2.43 
2.43 
2.43 

.878766 
.878656 

.878547 
.878438 
.878328 

.04 

.82 
.82  . 
.82 
.82 

.936866 
.937121 
.937377 
.937632 
.937887 

4  26 
4.26 
4.26 
4.25 
4.25 

.063134 
.062879 
.062623 
.062368 
.062113 

9 
8 
7 
6 
6 

66 

57 
58 
59 

.816361 
.816507 
.816652 
.816798 

2.43 
2.43 
2.43 
2.42 

.878219 
.878109 
.877999 
.877890 

.83 

.83 
.83 

.83 

.933142 
.938398 
.93*653 
.938908 

4.25 
4.25 

4.25 
4.25 

.061358 
.061602 
.061347 
.061092 

4 

3 

3 

60 

.816943 

2.42 

.877780 

1.83 

.939163 

4.25 

.060837 

0 

M. 

Cosine 

D.  1". 

Sine 

D.  1''. 

Cotang. 

D.I" 

Tang. 

M 

COSINES,  TANGENTS,  AND  COTANGENTS.        271 

410                                              138° 

M. 

Blue. 

D.I". 

Cosine. 

D  1". 

fcrng. 

D.1'. 

Ootaog. 

M. 

0 

a 

!817088 
.817233 

2.42 
2.42 

O  /1O 

9.877780 
.877670 
.877560 

1.83 
1.83 

IQQ 

9.939163 

.939418 
.939673 

4.26 
4.25 

A  OK 

0.060837 
.060582 
.060327 

60 
69 

68 

3 
4 

6 
6 

7 
8 
9 

.817379 
.817524 
.817668 
.817813 
.817958 
.818103 
.818247 

<6.4<6 

2.42 
2.42 
2.41 
2.41 
2.41 
2.41 
2.41 

.877450 
.877340 
.877230 
.877120 
.877010 
.876899 
.876789 

.OO 

1.83 
1.84 
1.84 
1.84 
1.84 
1.84 
1.84 

.939928 
.940183 
.940139 
.940694 
.940949 
.9412^ 
.941459 

4.<6W 

4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 

.060072 
.059817 
.059561 
.059306 
.059051 
.058796 
.058541 

67 
56 
55 
54 
53 
52 
61 

10 
11 
12 
13 
14 
15 
16 
17 
18 

9.818392 
.818536 
.818681 
.818825 
.818969 
.819113 
.819257 
.819401 
.819545 

2.41 

2.41 
2.40 
2.40 
2.40 
2.40 
2.40 
2.40 

9.876678 

.876568 
.876457 
.876347 
.876236 
.876125 
.876014 
.875904 
.876793 

1.84 

1.84 
1.84 

1.84 
1.85 
1.85 
1.85 
1.85 

9.941713 
.941968 
.942223 
.942478 
.942733 
.942988 
.943243 
.943498 
.943752 

4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 

0.058287 
.058032 
.057777 
.057522 
.057267 
.057012 
.056757 
.056502 
.056248 

60 
49 
48 
47 
46 
45 
44 
43 
42 

19 

.819689 

2.40 
2.39 

.875682 

1.85 
1.85 

.944007 

4^25 

.055993 

41 

20 
21 
22 
23 
24 
25 
26 
27 
28 

9  819832 
.819976 
.820120 
.820263 
.820406 
.820550 
.820693 
.820836 
.820979 

2.39 
2.39 
2.39 
2.39 
2.39 
2.39 
2.38 
2.38 

9.875571 
.875459 
.8753-18 
.875237 
.875126 
.875014 
.874903 
.874791 
.874680 

1.85 
1.85 

1.85 
1.86 
1.86 
1.86 
1.86 
1.86 

9.944262 
.944517 
.944771 

.945026 
.945281 
.945535 
.945790 
.946045 
.946299 

4.25 
4.25 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 

0.055738 
.055483 
.055229 
.054974 
.054719 
.054465 
.054210 
.053955 
.053701 

40 
39 
38 
37 
36 
35 
34 
33 
32 

29 

.821122 

2.38 
2.38 

.874568 

1.86 
1.86 

.946554 

4.24 

4.24 

.053446 

31 

30 
31 

9.821265 

.821407 

2.38 

9.874456 
.874344 

1.86 

9.946808 
.947063 

4.24 

0.053192 
.052937 

30 
29 

32 
33 
34 
35 
36 
37 
38 
39 

.821550 
.821693 
.821835 
.821977 
.822120 
.822262 
.822404 
.822546 

2.33 
2.33 
2.37 
2.37 
2.37 
2.37 
2.37 
2.37 
2.37 

.874232 
.874121 
.874009 
.873896 
.873784 
.873672 
.873560 
.873448 

1.86 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 

.947318 
.947572 
.947827 
.948081 
.948335 
.948590 
.948844 
.949099 

4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 

.052682 
.052428 
.052173 
.051919 
.051665 
.051410 
.051156 
.050901 

28 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.822688 
.822330 
.822972 
.823114 
.823255 
.823397 
.823539 
.823680 
.823821 
.823963 

2.37 
2.36 
2.36 
2.36 
2.36 
2.36 
2.36 
2.36 
2.35 

9.873335 
.873223 
.873110 

.872998 
.872885 
.872772 
.872659 
.872547 
.872434 
.872321 

1.87 
1.88 
1.88 
1.88 
1.88 
1.88 
1.88 
1.88 
1.88 

9.949353 

.949608 
.949862 
.950116 
.950371 
.950625 
.950879 
.951133 
.951388 
.951642 

4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 

0.050647 
.050392 
.050138 
.049884 
.049629 
.049375 
.049121 
.048867 
.048612 
.048358 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

2.35 

1.88 

4.24 

60 
61 
62 

9.824104 
.824245 
.824386 

2.35 
2.35 

9.872208 
.872095 
.871981 

1.89 
1.89 

9.951896 
.952150 
.952405 

4.24 
4.24 

0.048104 

.047850 
.047595 

10 
9 
8 

53 

.824527 

2.35 

.871868 

1.89 

.952659 

4.24 

.047341 

7 

54 
55 

.824668 
.824808 

2.35 
2.35 

.871755 
.871641 

1.89 

1.89 

.952913 
.953167 

4.24 
4.24 

.047087 
.046833 

6 

6 

56 
67 
63 
69 

.824949 
.825090 
.825230 
.825371 

2.34 
2.34 
2.34 
2.34 

.871528 
.871414 
.871301 
.871187 

1.89 
1.89 
1.89 
1.89 

.953421 
.953675 
.953929 

.954183 

4.24 
4.24 
4.23 
4.23 

.046579 
.046325 
.046071 
.045817 

4 

3 

a 
i 

60 

.825511 

2.34 

.871073 

1.90 

.954437 

4.23 

.045563 

0 

M. 

Cosine. 

D.I". 

Slue. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

M. 

48° 


.OGARITHMIC     SIXES, 


137* 


M. 

Sine. 

D.  1''. 

Codne. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

M. 

0 

2 
3 
4 
5 
6 
7 
8 
9 

9.8255  1  1 
.825651 
.825791 
.825931 
.826071 
.82621  1 
.826351 
.826491 
826631 
.826770 

2.34 
2.34 
2.33 
2.33 
2.33 
2.33 
2.33 
2.33 
2.33 
2.33 

9.871073 
.870960 
.870846 
.870732 
.870618 
.870504 
.870390 
.870276 
.870161 
.870047 

.90 

.90 
.90 
.90 
.90 
.90 
.90 
.90 
.91 
.91 

9.954437 
.954691 
.954946 
.955200 
.953454 
.9557(13 
.955%! 
.956215 
.956469 
.956723 

4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 

0.045563 
.045309 
.045054 
.044800 
.044546 
.044292 
.044039 
.043785 
.043531 
.043277 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

ie 
11 

9.826910 
.827049 

2.32 

O  '}•.) 

9.869933 
.869318 

.91 

9.956977 
.957231 

4.23 

A  OQ 

0.043023 
.042769 

50 

49 

12 
13 
14 
15 
16 
17 
18 

.827139 
.82732-3 
.827467 
.827606 

.827745 
.827884 
.828023 

A.&A 

2.32 
2.32 
2.32 
2.32 
2.32 
2.31 

O  Ql 

.869704 
.869589 
.869474 
.869360 
.869245 
.869130 
869015 

.91 
.91 
.91 
.91 
.91 
.91 
.92 

QO 

.957435 
.957739 
.957993 
.958247 
.958500 
.958754 
.959008 

4.-6O 

4.23 
4.23 
4.23 
4.23 
4.23 
4.23 

A  O1 

.042515 
.042261 
.042007 
.041753 
.041500 
.041246 
.040992 

48 
47 
46 
45 
44 
43 
42 

19 

.828162 

£.&{ 

2.31 

.868900 

,Wm 

.92 

.959262 

%JH 

4.23 

.040738 

41 

20 
21 

9.828301 
.828439 

2.31 

9.868785 
.863670 

.92 

9.959516 
.959769 

4.23 

0.040484 
.040231 

40 
3S 

22 

.823578 

2.31 

.868555 

.92 

.960023 

4.23 

.039977 

38 

23 
24 

.828716 

.828855 

2.31 
2.31 

.868440 
.868324 

.92 
.92 

.960277 
.960530 

4.23 
4.23 

.039723 
.039470 

37 

36 

25 
26 
27 
28 
29 

.828993 
.829131 
.829269 
.829407 
.829545 

2.31 
2.30 
2.30 
2.30 
2.30 
2.30 

.8GS209 
.863093 
.867978 
.867862 
.867747 

.92 
.92 
.93 
.93 
.93 
.93 

.960784 
.961033 
.961292 
.961545 
.961799 

4.23 
4.23 
4.23 
4.23 
4.23 
4.23 

.039216 
.03^962 
.038708 
.038455 
.038201 

35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 

9.829683 
.829821 
.829959 
.830097 
.830234 
830372 
.830509 

2.30 
230 
2.29 
2.29 
2.29 
2.29 

9.867631 
.867515 
.867399 
.867233 
.867167 
.867051 
.866935 

.93 
.93 
.93 
.93 
.93 
.94 

9.962052 
.962306 
.962560 
.962813 
.963067 
.963320 
.963574 

4.23 
4.23 
4.23 
4.23 
4.23 
4.23 

0.037948 
.037694 
.037440 
.037187 
.036933 
.036630 
.036426 

3C 

29 
28 
27 
26 

24 

37 
38 
39 

.83(1646 

.830784 
.830921 

2.29 
2.29 
2.29 
2.29 

.866819 
.866703 
.866586 

.94 
.94 
.94 
.94 

.963828 
.964081 
.964335 

4.23 
4.23 
4.23 
4.23 

.036172 
.035919 
.035665 

23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 

9.831058 
.831195 
.831332 
.831469 
.831606 
.831742 
.831879 
.832015 
.832152 

2.23 
2.2l? 

2.23 
2.28 
2.23 
2.28 
2.23 
2.27 

O  O7 

9.866470 
.866353 
.866237 
.866120 
.866004 
.865387 
.865770 
.865653 
.865536 

.94 
.94 
.94 
94 
.95 
.95 
.95 
.95 

nc 

9.964588 
.964842 
.965095 
.965349 
.965602 
.965855 
.966109 
.966362 
.966616 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

A  QO 

0.035412 
.035158 
.034905 
.034651 
.034393 
.034145 
.033891 
.033633 
.033384 

20 
19 
18 
17 
16 
15 
14 
13 
12 

49 

.832288 

<E»3S/ 

2.27 

.865419 

.yo 
.95 

.966869 

4MM 

4.22 

.033131 

11 

50 
51 

9.832425 

.832361 

2.27 

9.865302 

.865185 

.95 

9.967123 
.967376 

4.22 

0.032877 
.032624 

10 
9 

62 
63 
64 

.83-2697 
.832833 
.832969 

2.27 
2.27 
2.27 

.865063 
.864950 
.864833 

.95 
.95 
.96 

.967629 
.907,383 
.963136 

4.22 
422 
4.22 

.032371 
.032117 
.031864 

3 
7 
6 

66 

.833105 

2.27 

.864716 

.96 

.968389 

4.22 

.031611 

6 

66 
67 
68 
69 
60 

.833241 
.833377 
.833512 
.833648 
833783 

2.26 
2.28 

2.26 
2.26 
2.26 

.864593 
.86-14  SI 
.864363 
.864245 
.864127 

.96 
.96 
.96 
.96 
.96 

.968643 
.968896 
.969149 
.969403 
.969656 

4  22 
4.22 
4.22 
4.22 
4.22 

.031357 
.031104 
.030851 
.030597 
030344 

4 
3 
2 

0 

M. 

Cosine. 

D.  1". 

Siue. 

D.  1". 

Cotang. 

D.I'. 

Tang. 

M. 

132° 


4T* 


COSINES,  TANGENTS,  AND  COTANGENTS.        _Jio 

430                                                136" 

M. 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotaug. 

M. 

0 

1 
2 

9.833783 
.833919 
.834054 

2.26 
2.26 

9.864127 
.864010 
.883*92 

1.% 

1.97 

9.969656 
.969909 
.970162 

4.22 
4.22 
4  "22 

0.030344 
.030091 
.029838 

60 
59 

58 

3 

4 
5 
6 
7 
8 
9 

.834189 
.834325 
.834460 
.831595 
.834730 
.834865 
.834999 

2.25 
2.25 
225 
225 
2.25 
2.25 
2.25 
2.25 

.863774 
.86:1656 
.863538 
.863419 
.863301 
.863183 
.863064 

.97 
1.97 
1.97 
1.97 
1.97 
L97 
.97 
.97 

!  970669 
.970922 
.971175 
.971429 
.971682 
.971935 

4>22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

.029584 
.029331 
.029078 
.023825 
.023571 
.028318 
.023065 

57 
56 
55 
54 
53 
52 
51 

10 

9.835134 

9.86-2946 

9.972188 

0.027812 

50 

11 
12 
13 

14 
15 
16 
17 
18 
19 

.835269 

.8354(13 
.83.KJ.i8 
.835672 
.835807 
.835941 
.836075 
.836209 
.836343 

2.24 
2.24 
2.24 
2.24 
2.24 
2.24 
2.24 
2.23 
2.23 
2.23 

.862827 
.862709 
.862590 
.862471 
.862353 
.862234 
.862115 
.661996 
.861877 

^93 
.98 
.93 
.93 
1.93 
1.98 
1.93 
1.98 
1.99 

.972441 
.972695 
.972948 
.973201 
.973454 
.973707 
.973960 
.974213 
.974466 

4'22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

.027559 
.027305 
.027052 
.026799 
.026546 
.026293 
.026040 
.025787 
.025534 

49 
48 
47 
46 
45 
44 
43 
42 
41 

20 
21 

9.836477 
.836611 

2.23 

9.861758 
.861638 

.99 

9.974720 
.974973 

4.22 

A  *» 

0.025280 
.025027 

40 
39 

22 

1  23 
24 

.836745 
.836878 
.837012 

2.23 
2.23 
2.23 

.861519 
.861400 
.861280 

.99 
.99 
.99 

.975226 
.975479 
.975732 

4.  £& 
4.22 
4.22 

.024774 
.024521 
.024268 

38 
37 
36 

25 
26 

27 
28 
'29 

.837146 
.837279 
.837412 
.837546 
.837679 

2.23 
2.22 
2.22 
222 
2.22 
2.22 

.861161 
.861011 
.860922 

.860802 
.860632 

.99 
.99 
.99 
2.00 
2.00 
2.00 

.975985 
.976238 
.976491 
.976744 
.976997 

4>22 
4.22 
4.22 
4.22 
4.22 

.024015 
.023762 
.023509 
.023256 
.023003 

35 
34 
33 
32 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.837812 
.837945 
.838078 
.83821? 
.838344 
.838-177 
.838610 
.838742 
.838875 
.839007 

222 
222 
2.22 
2.21 
221 
2.21 
2.21 
2.21 
2.21 
2.21 

9.860562 
.86TH42 
.860322 
.860202 
.8600.32 
859962 
.859.342 
.859721 
.859601 
.859480 

2.00 
2.00 
2.00 
2.00 
2.00 
2.00 
2.01 
2.01 
2.01 
2.01 

9.977250 
.977503 
.977756 
.978009 
.978262 
.978515 
.978763 
.979021 
.979274 
.979527 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

0.022750 
022497 
.022244 
.021991 
.021738 
.021485 
.021232 
.020979 
.020726 
.020473 

30 
29 
23 
27 
26 
25 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.839440 
.839272 
.839404 
.839536 
.839668 
.839800 
.839932 
.840064 
.840196 
.84C328 

221 

2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.19 
2.19 

9.859360 
.859239 
.859119 
.858998 
.858877 
.858756 
.858635 
.858514 
.853393 
.858272 

2.01 
2.01 
2.01 
2.01 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 

9.979780 
.980033 
.980286 
.980538 
.980791 
.981044 
.981297 
.981550 
.981603 
.932056 

4.22 
4.22 
4.22 
4.22 
4.22 
4.21 
421 
4.21 
4.21 
4.21 

0.020220 
.019967 
.019714 
.019462 
.019209 
.018956 
.018703 
.018450 
.018197 
.017944 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

60 
61 
52 
63 
64 
65 
56 

9.840459 
.840591 

.840722 
.840.854 
.840985 
.841116 
.841-247 

2.19 
2.19 
219 
'.19 
2.19 
2.!  9 

9.858151 
.858029 

.857908 
.857786 
.857665 
.857543 
.85?422 

2.02 
2.02 
2.02 
2.03 
203 
203 

9.982309 
.982562 

[983067 
.9-^3320 
.9S3573 
.933826 

4.21 
421 
4.21 
421 
421 
421 

0.017691 
.017433 
.017186 
.016933 
.016680 
.016427 
.016174 

10 
9 

8 
7 

67 
58 
59 
60 

.841378 
.841509 
.841640 
.841771 

2.  18 
2.18 
2.18 
2.18 

.85?30() 
.857178 
.857056 
.856934 

2.03 
2.03 
2.03 
2.03 

.9S4U79 
.984332 
.984584 
.984837 

4.21 
4.21 
4.21 
4,21 

.01592! 
.015668 
.015416 
.015163 

0 

M. 

Coetra. 

D.  1". 

Bine. 

D.  1". 

Cotung. 

D.I". 

Tang. 

M. 

1330 


46° 


274 


TABLE    II.       LOGARITHMIC    SINES, 


M. 

Sine. 

D.  1". 

Cosine. 

D.  1". 

T»ng. 

D.  1". 

Cotang. 

M. 

7 
8 

9.841771 
.841902 
.842033 
.842163 
.842294 
.842424 
.842555 
.842685 
.842815 

2.13 
218 
218 
2.18 
2.17 
217 
217 
217 

9.856934 
.856812 
.856690 
.856568 
.856446 
.856323 
.856201 
.856078 
.855956 

203 
204 
204 
204 
204 
2.04 
2.04 
2.04 

9.984837 
.985090 
.985343 
.985596 

.985843 
.986101 
.986354 
.986607 
.936860 

4.21 
421 
4.21 
4.21 
4.21 
4.21 
421 
4.21 

0.015163 
.014910 
.014657 
.014404 
.014152 
.013899 
.013646 
.013393 
.013140 

60 
69 
68 
67 
58 
55 
54 
63 
52 

9 

.842946 

2.17 

.855833 

2.04 

.987112 

4.21 

.012388 

51 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

9.843076 
.843206 
.843336 
.843466 
.843595 
.843725 
.843355 
.843984 
.844114 
.844243 

2.17 
2.17 
216 
2.16 
216 
2.16 
216 
216 
216 
2.16 

9.855711 
.855538 
.855465 
.855342 
.855219 
.855096 
.854973 
.854850 
.854727 
.854603 

2.05 
2.05 
205 
205 
205 
205 
2.05 
2.05 
2.06 
2.06 

9.937365 
.987618 
.987871 
.988123 
.988376 
.988629 
.988882 
.989134 
.989387 
.989640 

4.21 
4.21 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

0.012635 
.012382 
.012129 
,011877 
.011624 
.011371 
.011118 
.010866 
.010613 
.010360 

60 
19 
48 
47 
46 
46 
44 
43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

9.844372 

.844502 
.844631 
.844760 
.844.889 
.845018 
.845147 
.845276 
.845405 
.845533 

2.15 
215 
215 
2.15 
215 
2  15 
215 
2  15 
2  14 
2.14 

9.854480 
.854356 
.854233 
.854109 
.853936 
.853362 
.853738 
.853614 
.853490 
.853366 

206 
206 
2.06 
2.06 
2.08 
206 
2.06 
207 
2.07 
2.07 

9.989393 
.990145 
.990393 
.990651 
.990903 
.991156 
.991409 
991662 
.991914 
.992167 

4.21 
421 
421 
4.21 
421 
4.21 
4.21 
4.21 
4.21 
4.21 

0.010107 
.009855 
.009602 
.009349 
.009097 
.008844 
.008591 
.008338 
.008088 
.007833 

40 
39 
38 
87 
36 
36 
34 
33 
31 
31 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

9.845662 
.845790 
.845919 
.846047 
.846175 
.846304 
.846132 
.846560 
.846688 
.846316 

2.14 
214 
214 
214 
214 
2.14 
213 
2.13 
213 
2.13 

9.853242 
.853118 
.852994 
.852369 
.852745 
.852620 
.852496 
.852371 
.852247 
.852122 

2.07 
2.07 
2.07 
2.07 
2.07 
2.08 
2.08 
2.08 
2.08 
2.03 

9.992420 
.992672 
.992925 
.993178 
.993431 
.993683 
.993936 
.994189 
.994441 
.994694 

4.21 
4.21 
4.21 
421 
4.21 
421 
421 
4.21 
4.21 
4.21 

0.007580 
.007328 
.007076 
.006,822 
.006569 
.006317 
.006064 
.005811 
.005559 
.005308 

30 
29 

28 
27 
26 
26 
24 
23 
22 
21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9.846944 

.847071 
.847199 
.847327 
.847454 
.847582 
.847709 
.847836 
.847964 
.843091 

2.13 
2.13 
2.13 
213 
2.12 
2.12 
2.12 
2.12 
2.1'2 
2.12 

9.851997 
.851872 
.851747 
.851622 
.851497 
.851372 
.851246 
.851121 
.ar>0996 
.850870 

2.08 
2.03 
2.08 
2.09 
2.09 
2.09 
209 
2.09 
2.09 
2.09 

9.994947 
.995199 
.995452 
.995705 
.995957 
.996210 
.996463 
.996715 
.996968 
.997221 

4.21 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

0.005053 
.004801 
.004548 
.004295 
.004043 
.003790 
.003537 
.003285 
.003032 
.002779 

20 
19 
18 
17 
16 
16 
14 
13 
19 
11 

60 
61 
52 
63 
64 
65 
68 
57 
58 
59 
60 

9.848218 
.848345 

.848472 
.848599 
.848726 
.848852 
.848979 
.849106 
.849232 
.849359 
.849485 

2.12 
2.12 
2.11 
2.11 
2.11 
2.11 
2.11 
2.11 
2.11 
2.11 

9.850745 
.850619 
.850493 
.850368 
.850242 
.8501  16 
.849990 
.849864 
.849738 
.84961  1 
.849485 

209 
2.10 
2.10 
2.10 
2  10 
2.10 
2.10 
2.10 
2.10 
2.11 

9.997473 
.997726 
.997979 
.998231 
.998484 
.998737 
.998989 
.999242 
.999495 
.999747 
0.000000 

4.21 
4.21 
4.21 
4.21 
421 
4.21 
4.21 
4.21 
4.21 
4.21 

0.002527 
.002274 
.002021 
.001769 
.001516 
.001263 
.001011 
.000758 
.000505 
.000253 
.000000 

10 
9 
8 
7 
6 
6 
4 

9 

1 
0 

M. 

Ooelne. 

D.1". 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

M. 

134° 


TABLE  III. 
NATURAL    SINES   AND    COSINES. 


275 


276 


TABLK    III         NATURAL     SIXES     AND    COSTXES. 


GO 

10 

30 

30 

4o 

M 

Sine! 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Coslu 

M 

0 

.00000 

One" 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

1J9756 

80 

1 

.000:29 

One. 

.01774 

.999.84 

03519 

.99938 

.05263 

.99>61 

.070115 

.99754 

50 

2 

00058 

One. 

.01803 

.999S4 

.03548 

.99937 

.05292 

.99-560 

.07034 

.99752 

58 

.  3 

.OUD87 

One. 

.01832 

.99983 

.03577 

.99936 

05321 

.99858 

.07063 

.99750 

37 

4 

.00116 

One. 

.01862 

.99983 

.036116 

.99935 

.05350 

.99857 

.07092 

.99748 

56 

5 

00145 

One. 

.01891 

.99982 

.03635 

99934 

.05379 

.99855 

.07121 

.99746 

56 

6 

.00175 

One. 

.01920 

.99932 

.(^664 

.99933 

.05408 

.99854 

.07150 

.99744 

54 

7 

00201 

One. 

.01949 

.99981 

.03093 

.99932 

.05437 

.99852 

.07179 

.99742 

53 

8 

00233 

One. 

.01978 

.99980 

.03723 

.99931 

.05466 

.99851 

.07208 

.99740 

52 

9 

00*62 

One. 

.02007 

.99980 

.03752 

.9993U 

.05495 

.99849 

07237 

.99738 

51 

10 

00291 

One. 

.02036 

.99979 

.03781 

.99929 

.05524 

.99847 

.07266 

.99736 

50 

11 

00320 

.99999 

.02065 

.99979 

.03-510 

.99927 

.05553 

.99846 

.07295 

.99734 

49 

12 

00341) 

.99999 

.02094 

.99978 

.03839 

.99926 

.05582 

.99844 

.07324 

.99731 

48 

13 

.00378 

.99999 

.02123 

.99977 

.03368 

.99925 

.0561  1 

.99842 

.07353 

.99729 

47 

14 

00407 

.99999 

.02152 

.99977 

.03897 

.99924 

.05640 

.99,841 

.07382 

.99727 

46 

15 

.00436 

.99999 

.02181 

.99976 

.03926 

.99923 

.05669 

.99839 

.07411 

.99725 

45 

16 

.0046--, 

.99999 

.02211 

.99976 

.03955 

.99922 

.05698 

.99838 

.07440 

.99723 

44 

17 

.00495 

.99999 

.0224'» 

.,j*76 

.03984 

.99921 

.05727 

.99336 

.07469 

.99721 

43 

18 

.00524 

.99999 

.0226y 

.00974 

|.  0401  3 

.99919 

.U5756 

.99834 

.07498 

.99719 

42 

19 

.00553 

.99993 

.02298 

.99974 

.04042 

.99918 

.05785 

.99833 

.07527 

.99716 

41 

20 

.00532 

.99993 

.02327 

.99973 

.04071 

.99917 

.05814 

.99831 

.07556 

.99714 

40 

21 

.00611 

.99998 

.02356 

.99972 

.04100 

.99916 

.05844 

.99329 

07585 

.99712 

39 

22 

.00640 

.99998 

.02385 

.99972 

.04129 

.99915 

.05873 

.99827 

07614 

.99710 

38 

23 

00669 

.99993 

.02414 

.99971 

.04159 

.99913 

.05902 

.99326 

07643 

.99708 

37 

24 

.00698 

99998 

.02443 

.99970 

.04188 

.99912 

.05931 

.99824 

07672 

.99705 

36 

25 

.00727 

.99997 

.02472 

.99909 

.04217 

.99911 

.05960 

.99822 

07701 

.99703 

35 

26 

.00756 

.99997 

.  02501 

.99969 

.04246 

.99910 

.05939 

.99821 

.07730 

.99701 

34 

27 

.00785 

.99997 

.02530 

.99963 

.04275 

.99909 

06018 

.99819 

.07759 

.99699 

33 

28 

.00814 

.999-7 

.02560 

.99967 

.0430-1 

999i  17 

.06047 

.99817 

.07788 

.99696 

32 

29 

.00844 

99W6 

.02589 

.99966 

.01333 

.99906 

.06076 

.99815 

.07817 

.99694 

31 

30 

OaS73 

.90996 

.02618 

.99966 

.04362 

.99905 

.06105 

.99813 

07846 

9%92 

30 

31 

oiwa 

.99996 

.02647 

99965 

.04391 

.99904 

.06134 

.99812 

07875 

.9Sti8i> 

•29 

32 

.00931 

.99996 

.02676 

.99964 

.04420 

.99902 

.06163 

.99310 

07904 

.99687 

IS 

33 

.00960 

.99995 

.02705 

.99963 

.04449 

.99901 

.06192 

.99808 

07933 

.99685 

17 

34 

.00939 

.99995 

.02734 

.99963 

.04478 

.99900 

.06221 

.99806 

07962 

.99683 

26 

35 

.01018 

.99995 

.02763 

.99962 

.04.307 

.99898 

.06251* 

.99804 

.07991 

.99680 

25 

36 

.01047 

99995 

.02792 

.99961 

.04536 

.99897 

.06279 

.99303 

.08020 

.99678 

24 

37 

.01076 

99994 

02821 

.99960 

.04565 

.99396 

.06308 

.99301 

.08049 

.99676 

23 

38 

.01105 

.99994 

02350 

.99959 

.04594 

.99894 

.06337 

.99799 

.03078 

.9%73 

22 

39 

.01134 

.99994 

02879 

.99959 

.04623 

.99893 

.06366 

.99797 

.08107 

.99671 

21 

40 

01164 

.99993 

02908 

.99958 

.04653 

.99892 

.06395 

.99795 

.08136 

.99668 

20 

41 

.01193 

.99993 

.02938 

.99957 

.046812 

.99890 

.06124 

.99793 

.08165 

.99666 

19 

42 

.01222 

.99993 

02967 

.99956 

.04711 

.99889 

.06453 

.99792 

.08194 

.99664 

18 

43 

.01251 

.99992 

.02996 

.99955 

.04740 

.99888 

.06482 

.99790 

.08223 

.99661 

17 

44 

.01280 

.99992 

.03025 

.99954 

.04769 

.99886 

06511 

.99788 

08252 

.99659 

16 

45 

.01309 

.99991 

.03054 

.99953 

.04798 

.99885 

.06540 

.99786 

.08281 

99657 

15 

46 

.01338 

.99991 

03083 

99952 

.04827 

.99883 

.06569 

.99784 

.08310 

.99654 

14 

47 

.01367 

.99991 

.03112 

.99952 

.04856 

.99882 

.06598 

.99782 

.08330 

.99652 

13 

48 

.01396 

.99990 

.03141 

.99951 

.04885 

.99881 

.06627 

.99780 

.08368 

.99649 

12 

49 

.01425 

.99990 

.03170 

.99950 

.04914 

.99879 

.06656 

.99778 

.08397 

.99647 

11 

50 

.01454 

.999S9 

.03199 

.99949 

.04943 

.99378 

.06685 

.99776 

.08426 

.99644 

10 

51 

.01483 

.99989 

.03228 

.99948 

.04972 

.99876 

.06714 

.99774 

08455 

.99642 

9 

52 

.01513 

.99989 

.03257 

.99947 

.051(01 

.99875 

.06743 

.99772 

08484 

.99639 

8 

53 

.01542 

.99988 

03286 

.99946 

.05030 

.99873 

.06773 

.99770 

08513 

.99637 

7 

54 

01571 

.999-18 

.03316 

.99945 

.05059 

.99872 

.06802 

.9976^ 

.03542 

.99635 

6 

55 

.01600 

.99937 

03345 

.99944 

.05088 

.99870 

.06831 

.99766 

08571 

.99632 

5 

56 

01629 

.99937 

.03374 

.99943 

.05117 

.99369 

.08860 

.99764 

.086r)0 

.99630 

4 

57 

01658 

.99986 

.03403 

.99942 

.05146 

.99*67 

.06389 

.99762 

.08629 

.99627 

3 

58 

.01687 

.99986 

.03432 

.99941 

.05175 

.99866 

.06918 

.99760 

08658 

.99625 

2 

59 

01716 

.99985 

.03461 

.99940 

.05205 

.99864 

.06947 

.99758 

.08687 

.99622 

60 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

.99756 

.08716 

.99619 

0 

M: 

Ooein. 

Slue 

Cosin. 

Sine. 

Cosin! 

Sine. 

Cosin. 

Sine. 

Coein. 

Sine. 

M. 



89° 

883 

87° 

86° 

850 

TABLE    III.        NATURAL     SIXES     AND    COSINES. 


50 

GO 

70 

8° 

9° 

M 

Sine. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Couin. 

M. 

0 

08716 

.99619 

.1O453 

.99452 

.12137 

.99255 

.13917 

.99027 

.1564c 

1)8769 

60 

1 

.0874') 

.99617 

.10482 

.994*  9 

.12216 

.99251 

.13946 

.99023 

.15672 

98764 

59 

2 

08774 

.99614 

10511 

.99446 

.12245 

.99.^43 

.13975 

.99019 

.15701 

.93760 

58 

3 

.08*13 

99612 

.  10.i4<) 

.9944.1 

.  12274 

.99244 

.14004 

.99015 

.15730 

.98755 

57 

4 

.08831 

99&I9 

10569 

.99440 

.12302 

.99240 

.14033 

.99011 

.15753 

.98751 

56 

5 

.08860 

99607 

10597 

.09437 

.12331 

.99237 

.14061 

.99006 

.15787 

.98746 

56 

6 

0888« 

99rt.ll 

10626 

.J..134 

.12360 

.99233 

.14090 

.99002 

15816 

.98741 

54 

7 

08918 

.996(12 

10655 

.99431 

.12389 

.99230 

.14119 

9899s 

.15,845 

.98737 

53 

6 

08947 

.99599 

10684 

99428 

.12418 

.99226 

.14143 

.98994 

.15873 

.98732 

52 

9 

.08976 

.99596 

10713 

.91)121 

.12447 

.99222 

.14177 

.98990 

.15902 

.98728 

51 

10 

.09(105 

.99594 

10742 

.99421 

.12476 

.99219 

.14205 

.98986 

.15931 

.98723 

50 

11 

09034 

.99591 

10771 

.99418 

.12504 

.99215 

.14234 

.98932 

.15959 

.98718 

49 

12 

09063 

.9953- 

10800 

.99415 

.12533 

.99211 

.14263 

.98978 

.15938 

.98714 

48 

13 

.09092 

.99586 

10829 

.99112 

.12562 

.99208 

.14292 

.98973 

.16017 

.98709 

47 

14 

09121 

.9958:5 

10358 

.99409 

.12591 

.9920-1 

.143-20 

.98969 

16046 

.98704 

46 

15 

.09150 

.99580 

.10*87 

.99406 

.12620 

.99200 

.14349 

.98965 

.16074 

.98700 

45 

16 

.09179 

.99573 

10916 

.99402 

.12649 

.99197 

.14378 

.93961 

.16103 

.98695 

44 

.09208 

.99575 

10945 

.99399 

.12678 

.99193 

.14407 

.98957 

16132 

.98690 

43 

18 

(19237 

.99572 

10973 

.99396 

.12706 

.99189 

.14436 

.98953 

16160 

.98686 

42 

19 

09266 

99570 

11002 

.99393 

.12735 

.991,86 

.14464 

.98WS 

161  89 

:  98681 

41 

20 

0929;) 

.99567 

1  1031 

.99390 

.12764 

.99182 

.14493 

.98944 

16218 

.98676 

40 

21 

09324 

.99564' 

11060 

.993-6 

.12793 

.99178 

.14522 

.98940 

.16246 

.9,8671 

39 

2*2 

09353 

.99562 

1  1089 

99383 

.12822 

.99175 

.14551 

.98936 

16275 

.98667 

38 

23 

09382 

•99559 

11118 

.99380 

.12851 

.99171 

.14580 

.98931 

16304 

.98662 

37 

24 

09111 

.99556 

i!147 

.99377 

.12880 

.99167 

.  14603 

.98927 

16333 

.98657 

36 

25 

09440 

.99553 

1  1176 

.99374 

.1290S 

.99163 

.14637 

.98923 

16361 

.98652 

35 

26 

09469 

.99551 

.11205 

99370 

.12937 

.99160 

14666 

98919 

16390 

.98643 

34 

27 

09498 

.99548 

.I12:« 

.99367 

.12966 

.99156 

.14695 

98914 

16419 

.98643 

33 

28 

095-27 

.99545 

11263 

.99364 

12995 

99152 

14723 

.98910 

16447 

.98638 

32 

29 

(19556 

99542 

11291 

.99360 

.13024 

.99148 

14752 

.98906 

16476 

.98633 

31 

30 

09585 

.99540 

11320 

.99357 

13053 

.99144 

.14781 

.98902 

.  16505 

.98629 

30 

31 

.09614 

.99537 

11349 

99354 

.13031 

.99141 

14810 

.98897 

16533 

.98624 

29 

32 

(19642 

.99534 

11378 

.99351 

.13110 

.99137 

.14838 

.98893 

16562 

.98619 

28 

33 

(19671 

.99531 

.11407 

.99347 

.13139 

.99133 

.14867 

98889 

16591 

.98614 

27 

34 

.09700 

.9952S 

11436 

99344 

13168 

.99129 

.  14896 

.98834 

16620 

.98609 

26 

35 

(19729 

.99526 

11465 

99341 

.13197 

.99125 

.14925 

.98830 

16643 

.93604 

25 

36 

09758 

.99523 

11494 

99337 

.13226 

.99122 

14954 

.98876 

16677 

.98600 

24 

37 

1)9787 

.99520 

11523 

.99334 

.13254 

.99118 

14982 

.98871 

16706 

93595 

23 

3« 

09816 

99517 

11552 

99331 

.13283 

.99114 

.15011 

.98867 

16734 

.98590 

22 

39 

09^45 

.99514 

.11580 

99327 

.13312 

.99110 

15040 

.98863 

16763 

98585 

21 

40 

.093*4 

.99511 

11609 

.99324 

.13341 

.99106 

15069 

.98853 

16792 

.98530 

20 

41 

09903 

.99508 

.11638' 

.99320 

.13370 

.99102 

.15097 

.98854 

16820 

.98575 

19 

42 

.09932 

.99506 

11667 

.99317 

.13399 

.99098 

15126 

.98849 

16849 

.98570 

18 

43 

.09961 

.99503 

.11696 

.99314 

.  13427 

.99094 

15155 

.98845 

16378 

.98565 

17 

44 

.09990 

.99500 

.11725 

.99310 

.13456 

.99091 

15184 

.98841 

16906 

.98561 

16 

45 

.10019 

.99497 

.11754 

.99307 

.13485 

.99087 

.15212 

.98836 

.16935 

.98556 

15 

46 

.10048 

99494 

.11783 

.99303 

.13514 

.99083 

15241 

.98332 

16964 

.98551 

14 

47 

10077 

99491 

.11812 

.99300 

.13543 

.99079 

.15270 

.98327 

16992 

.98546 

13 

48 

10106 

99488 

.11840 

.99297 

.13572 

.99075 

.15299 

.98323 

17021 

.98541 

12 

49 

10135 

.99485 

11869 

.99293 

.13600 

.99071 

.15327 

.98818 

.17050 

.98536 

11 

50 

.10164 

994*2 

.11898 

.99290 

.13629 

.99067 

.15356 

.98814 

.17078 

.98531 

10 

5i 

10192 

99479 

.11927 

.99286 

.13658 

.99063 

.15385 

.93309 

.17107 

.98526 

9 

52 

10*21 

.99476 

11956 

.99283 

.13687 

.99059 

.15414 

.98305 

.17136 

.93521 

8 

53 

10250 

99473 

1  1985 

.99279 

13716 

.99055 

.15442 

.98800 

.17164 

.98516 

7 

54 

KI279 

.99470 

12014 

.99276 

.13744  .99051 

.15471 

.98796 

.17193 

.9^*511 

6 

55 

10318 

99467 

12043 

.99272 

13773  .99047 

15500 

.98791 

.17222 

98506 

5 

56 

111337 

99164 

12071 

.99269 

13802 

.99(^3 

15529 

.98787 

.17250 

.98501 

4 

57 

10366 

.99461 

12100 

.99265 

13831 

.99039 

15557 

.98782 

.17279 

98496 

3 

53 

IK395 

.994.V 

12129 

99-262 

13860 

.99035 

15586 

.98778 

17303 

.98491 

2 

59 

10424 

.99455 

12153 

.99258 

13839 

.99(131 

15615 

.98773 

17336 

9*486 

1 

60 

10453 

99452 

12187 

.99255 

13917 

.99027 

15643 

.98769 

.17365 

.9M431 

0 

M: 

C^IiT 

Sine. 

Cosin. 

Sine. 

Costa. 

SineT 

CoainT 

Sine. 

Cosin 

Sin*. 

ML 

8*0 

830 

83° 

810 

800 

278 


TABLE    III.        NATURAL    SINES     ANI1    COSINES. 


100 

110 

120 

130 

14* 

M. 

Sb». 

Corfn- 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Coain. 

M. 

~0 

17365 

.98481 

19081 

.9*163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

60 

I 

17393 

.9>476 

19109 

.98157 

20.-S20 

.97809 

.2*333 

.97430 

.24220 

97023 

59 

2 

17422 

.98471 

.19138 

.98152 

.20S48 

.97-<o:j 

22552 

.97424 

24249 

.97015 

58 

3 

17451 

.98466 

IU  167 

.9.8146 

.211877 

.97797 

.22ASO 

.97417 

24277 

.97(KI8 

57 

4 

.17479 

.98461 

.19195 

.98140 

.20905 

.97791 

.22608 

.97411 

24305 

.97001 

56 

5 

17508 

.9-8455 

19224 

.98135 

.20933 

.977.84 

.22637 

.97404 

.24333 

.96994 

55 

6 

17537 

.98450 

19252 

.98129 

20962 

.97778 

226(55 

.97398 

.24362 

.969»7 

54 

7 

.17565 

98445 

19281 

.98124 

.20990 

.97772 

.22693 

97391 

24390 

.96980 

53 

8 

17594 

.9844(1 

.I93i  »9 

.98118 

.21019 

.97766 

22722 

973.84 

.24418 

96973 

52 

9 

.17623 

.98435 

1933-8 

.981  12 

.2HM7 

.97760 

22750 

.97378 

.24446 

.96966 

51 

10 

.17651 

.98430 

19366 

.98107 

.21076 

.97754 

.22778 

.97371 

24474 

.96959 

50 

11 

.17680 

.9S425 

19395 

.9SI01 

.21104 

.97748 

.22807 

.97365 

.24503 

.96952 

49 

12 

.17708 

.98420 

19123 

.98096 

21132 

.97742 

.22835 

97358 

.24531 

.96945 

48 

13 

.17737 

.98414 

19452 

.98090 

21161 

.97735 

22863 

.97351 

24559 

.96937 

47 

14 

.17766 

.9.8409 

19481 

.93084 

.21189 

.97729 

.88392 

97345 

.24587 

.96930 

46 

15 

.17794 

.984.H 

.19509 

.98079 

.21218 

.97723 

22920 

.97338 

.24615 

.96923 

45 

16 

.17823 

.98399 

.19538 

.98073 

212-46 

.97717 

22948 

.97331 

24644 

.96916 

44 

17 

.17852 

98394 

19566 

.98067 

.21275 

.97711 

22977 

.97325 

21672 

.96909 

43 

IS 

.17880 

.98389 

.  19595 

.9*  "61 

.21303 

.97705 

23005 

.97318 

.24700 

.96902 

42 

19 

.17909 

98383 

I9&23 

.9.8056 

.21331 

.976SH 

23033 

.97311 

24728 

96,894 

41 

20 

.17937 

.98378 

19652 

.98050 

.21360 

.97692 

.23062 

.97304 

24756 

.96887 

40 

21 

17966 

.98373 

19680 

.98044 

.21388 

.976^6 

23090 

.9729-} 

24784 

.96880 

39 

22 

17995 

.98368 

19709 

.98039 

21417 

.97680 

23118 

97291 

24813 

96873 

38 

23 

18023 

.98362 

19737 

98033 

.21445 

.97673 

23146 

972^4 

24841 

.96S66 

37 

24 

1H052 

.98357 

19766 

.98027 

21474 

.97667 

.23175 

.97278 

24S69 

.96858 

36 

25 

18081 

.98352 

19794 

.98021 

.215(12 

.97661 

232'  13 

.97271 

24*97 

.96*51 

M 

26 

I81(»9 

.98347 

19823 

98016 

.21530 

.97655 

23231 

.97261 

24925 

96844 

34 

27 

18133 

.98341 

19851 

.98010 

21559 

.9764* 

.23260 

.97257 

24954 

96837 

33 

28 

18166 

.9.8336 

19880 

.98001 

.21587 

.97642 

23288 

.97251 

24982 

96829 

32 

29 

18195 

98331 

1990S 

97998 

.21616 

97636 

233  16 

.97241 

.25010 

96-i22 

31 

30 

.18224 

.93305 

.19937 

.97992 

.21644 

.97630 

23345 

.97237 

25038 

96815 

30 

31 

18252 

98320 

19965 

.97987 

.21672 

.97623 

23373 

97210 

.25066 

.96.807 

29 

32 

18281 

.98315 

19994 

.97981 

.21701 

.97617 

23401 

97223 

.25094 

.96800 

28 

33 

18309 

983|(» 

20022 

97-J75 

21729 

.9761  1 

23429 

97217 

.25122 

96793 

27 

34 

18338 

.98304 

20051 

.97969 

21758 

.976m 

23458 

97210 

25151 

96786 

26 

35 

18367 

.98299 

20079 

.97963 

21786 

.97598 

23486 

.97203 

25179 

96778 

25 

36 

18395 

.98294 

2'M08 

.97958 

.21814 

.97592 

23514 

97196 

.25207 

.96771 

24 

37 

18424 

.98288 

20136 

.97952 

.21843 

.97585 

23542 

97189 

.28235 

96764 

23 

38 

1^452 

.9.8283 

20165 

.97946 

21871 

.97579 

23571 

.971H2 

.25263 

96756 

22 

39 

18481 

.98277 

20193 

97940 

21899 

.97573 

2«99 

97176 

.25291 

.96749 

21 

40 

.18509 

98272 

20222 

97934 

.21928 

.97566 

23627 

97169 

.2532(1 

.96742 

20 

41 

.18538 

.9.8267 

.area) 

97928 

21956 

.97560 

.23656 

.97162 

25348 

.96734 

19 

42 

.18567 

.98261 

20279 

.97922 

.21985 

.97553 

23684 

.97155 

.25376 

.96727 

18 

43 

18595 

.98256 

20307 

.97916 

.22013 

.97547 

23712 

.97148 

25404 

.96719 

17 

44 

13624 

.98250 

20336 

97910 

.22041 

.97541 

23740 

.97141 

25432 

.96712 

16 

45 

.18652 

.98245 

20364 

97905 

.22070 

.97534 

23769 

.97134 

25460 

96705 

15 

46 

.18681 

.98240 

20393 

97899 

.22098 

.97528 

.23797 

.97127 

25488 

.96697 

14 

47 

.18710 

.98234 

.20421 

.97893 

22126 

.97521 

23825 

.97120 

.25516 

.96690 

13 

48 

18738 

.982-29 

20450 

.97887 

.22155 

.97515 

.23853 

97113 

25545 

.966.82 

12 

49 

18767 

.98223 

.20478 

.97881 

.22183 

.97508 

23882 

.97106 

25573 

.96675 

11 

50 

18795 

98218 

.20507 

.97875 

.22212 

.97502 

23910 

.97100 

25601 

96R67 

10 

51 

18824 

.98212 

20535 

.97869 

22240 

.97496 

23938 

97093 

25629 

96660 

9 

52 

18852 

982(17 

.20563 

.97863 

22268 

.974S9 

23'.)66 

.97086 

25657 

96653 

8 

53 

18881 

98201 

.20592 

.97857 

22297 

.97483 

23995 

.97079 

2nfW5 

.9ffR45 

7 

54 

18910 

98196 

.2'i620 

.97851 

22325 

.97476 

24023 

97072 

25713 

.9663,8 

6 

55 

18938 

98190 

20649 

.97845 

223531.97470 

.24051 

97065 

25741 

96630 

5 

56 

18967 

.98185 

20677 

.97839 

22382  |.  97463 

.24079 

9705* 

25769 

.96623 

4 

57 

18995 

.98179 

20706 

.97833 

22410  .97457 

.24108 

.97051 

•25798 

.96615 

3 

58 

19(124 

.98174 

20734 

.97827 

22438  .97450 

.24136 

97044 

25826 

.96608 

2 

59 

19052 

.98168 

20763 

.97-i21 

22467|.97444 

.24164 

.97037 

25854 

.96600 

1 

60 

I90S1 

.98163 

.20791 

.97815 

224951.97437 

24192 

97030 

25882 

96593 

0 

M. 

Coeiu 

Sine. 

Cosin. 

Sine. 

Cosin.  Sine. 

Cosir  . 

Sine. 

Cosin. 

Sice. 

s: 

TOO 

780 

770 

76° 

T50 

TABLE    III.        NATURAL    SINES    AND    COSINES. 


279 


150 

160 

170 

18° 

19° 

M 

Sloe. 

Cosin 

Sine. 

Cosin 

Sine. 

Cosin. 

Sine. 

Ccsin. 

Slno. 

Co«in. 

M. 

0 

.25382 

^6593 

^27564 

"96126 

.29237 

."95630 

730902 

95106 

.32557 

94552 

60 

1 

.25910 

.96585 

.27592 

961)3 

.29*85 

.956*2 

.3H929 

95097 

.32584 

94542 

59 

2 

.25933 

.96578 

.27620 

96110 

.29293 

.95613 

.30957 

.95088 

.32612 

94533 

58 

3 

.25966 

.96570 

.27648 

96102 

.2-J321 

95605 

.30985 

.95079 

.32639 

94523 

57 

4 

.25994 

.96562 

.27676 

96094 

.29343 

.95596 

.31012 

.95070 

.32667 

.94514 

56 

5 

.26022 

.96555 

.87704 

960^6 

.29376 

.955^8 

.31040 

.95061 

.32694 

.94504 

55 

6 

.26050 

.96547 

27731 

96073 

.29404 

.95579 

.31063 

.95052 

.32722 

.94495 

54 

7 

.26079 

.96340 

.27759 

96070 

.211432 

.95571 

.31095 

.95043 

.32749 

.94485 

53 

8 

.26107 

.96532 

.27787 

96062 

.29460 

.95562 

.31123 

.95033 

32777 

.94476 

52 

9 

.26135 

.96524 

.27315 

96054 

.29487 

.95554 

.31151 

.95024 

.32804 

.94466 

51 

10 

26163 

96517 

.27843 

96046 

.29515 

.95545 

.31173 

.95015 

.32832 

.94457 

50 

11 

26191 

.96509 

.27371 

.96037 

.29.">43 

.95536 

.31206 

.95006 

.32859 

.94447 

49 

12 

.26219 

.96502 

.27899 

96029 

.29571 

.9552S 

.31233 

.94997 

.32887 

.94438 

48 

13 

.26247 

.96494 

.27927 

.96021 

.29.-i99 

.95519 

.31261 

.94933 

.32914 

.94428 

47 

14 

.26275 

.964  -iff 

.27955 

.96013 

.29626 

.95511 

.31239 

.94979 

.32942 

.94418 

46 

15 

.26303 

.96479 

.27983 

.96005 

.29654 

.95502 

.31316 

.94970 

.32969 

.94409 

45 

16 

26331 

.96471 

.23011 

.95997 

.29632 

.95493 

.31344 

.94961 

.32997 

.94399 

44 

17 

26359 

.96463 

.2.3039 

959-0 

.29710 

.95485 

.31372 

.94952 

.33024 

.94390 

43 

IS 

26337 

.96456 

.23067 

.95931 

.29737 

.95476 

.31399 

.94943 

.33051 

.94380 

42 

19 

.26115 

96448 

.23095 

.95972 

.29765 

.95467 

.31427 

.94333 

.33079 

.94370 

41 

20 

26443 

.96440 

.28123 

.95964 

.29793 

.95459 

.31454 

.94924 

.33106 

.94361 

40 

21 

26471 

.96433 

.28150 

.95956 

.29821 

.95450 

.31482 

.94915 

.33134 

94351 

39 

22 

.26500 

.96425 

.28178 

.95943 

.29849 

.95441 

.31510 

.94906 

33161 

.94342 

33 

23 

.26523 

.96417 

.28206 

.95940 

.29876 

.95433 

.31537 

.94897 

33189 

.94332 

37 

24 

.26556 

.96410 

.23234 

.95931 

.29904 

95424 

.31565 

.94838 

33216 

.94322 

36 

25 

265S4 

.964i  12 

.28262 

.95923 

.29932 

.95415 

.31593 

.94878 

33244 

.94313  35 

26 

26612 

.96394 

.88290 

.95915 

.29960 

.95407 

.31620 

.94869 

33271 

.94303  34 

27 

26640 

.963s6 

.23318 

.95907 

.29987 

.95398 

.31648 

.94860 

33298 

.94*93 

33 

28 

26668 

.96379 

.28346 

.95393 

.30015 

.95389 

.31675 

.94851 

33326 

.94284 

32 

29 

26696 

96371 

2.3374 

.95890 

.30043 

.95330 

.31703 

94842 

33353 

.94274!  31 

30 

26724 

.96363 

.28402 

.95882 

.30071 

.95372 

.31730 

.94832 

33381 

.94264 

30 

31 

26752 

.96355 

.28429 

.95874 

.30098 

.95363 

.31758 

.94323 

33408 

.94254 

29 

32 

26780 

.96347 

23457 

.95,365 

.30126 

.95354 

.31786 

.94814 

33436 

.94245 

28 

33 

26308 

.96340 

23135 

.95357 

.30154 

.95345 

.31813 

.94805 

33463 

.94*35.  27 

34 

26836 

.963.52 

23513 

.95349 

.30182 

.95337 

.31841 

.94795 

£3490 

.94225 

26 

35 

26364 

.96324 

28541 

.95841 

.30209 

.95323 

.31863 

.94786 

33518 

.94215 

25 

36 

26392 

.96316 

28569 

.95332 

30237 

.95319 

.31396 

.94777 

33545 

.94206 

24 

37 

26920 

.96308 

.23597 

.95324 

30265 

.95310 

.31923 

.94763 

33573 

94196 

23 

38 

26943 

.96301 

23625 

.95316 

.30292 

.95301 

.31951 

.94758 

.33600 

.94186 

22 

39 

26976 

96-293 

28652 

.95307 

.30320 

.95293 

.31979 

.94749 

33627 

.94176121 

40 

27004 

.96235 

23630 

.95799 

.30348 

.95284 

.32006 

.94740 

33655 

.94167 

20 

41 

.27032 

96277 

23708 

.95791 

.30376 

.95275 

.32034 

.94730 

33682 

.94157 

19 

42 

.27060 

.96269 

23736 

.95782 

.30403 

.95266 

.32061 

.94721 

33710 

.94147 

18 

43 

.27088 

96261 

23764 

.95774 

.30431 

.95257 

.32039 

.94712 

.33737 

.94137 

17 

44 

27116 

.96253 

28792 

.95766 

.30459 

.95248 

.32116 

.94702 

.33764 

.94127 

16 

45 

.27144 

.96246 

.28820 

.95757 

.30486 

.95240 

.32144 

.94693 

33792 

.94118 

15 

46 

27172 

.96233 

.28847 

.95749 

.30514 

.95231 

.32171 

.94684 

33819 

.94108 

(4 

47 

27200 

.96230 

.28375 

.95740 

.3'  1542 

.95222 

.32199 

.94674 

.33846 

.94098 

13 

43 

27228 

.913222 

23903 

.95732 

.30570 

.95213 

.32227 

.94665 

33874 

.94088 

12 

49 

27256 

.96214 

.28931 

.957*4 

.30597 

.95204 

32254 

.94656 

33901 

.94078 

II 

50 

27*34 

.96206 

.23959 

.95715 

.30625 

.95195 

.32282 

.94646 

33929 

.94068 

10 

61 

27312 

.96193 

.28937 

.93707 

.30653 

.95186 

.32309 

.94637 

.33956 

.94058 

9 

52 

27340 

.96.90 

.29015 

.95693 

.30680 

.95177 

.32337 

.94627 

.33983 

.94049 

8 

53 

27368 

.96182 

.29042 

.95690 

.30708 

.95163 

.32364 

.94618 

.34011 

.94039 

7 

54 

27396 

96174 

.29070 

.95631 

.30736 

.95159 

32392 

.94609 

.34033 

.94029 

6 

55 

27124 

96166 

.29099 

.95673 

.30763 

.95150 

32419 

.94599 

.34065 

.94019 

6 

56 

27452 

96158 

.29126 

.95664 

30791  .95142 

32447 

.94590 

.340931.94009 

4 

57 

27430 

.96150 

29154 

.95656 

30819  .95133 

.32474 

.94580 

.34120 

.93999 

3 

58 

.27508 

.96142 

29182 

.95647 

.30346  .95124 

.32502 

.94571 

.34147 

.939^9 

2 

59 

27536 

.96134 

.29209 

.95639 

.30874  .95115 

.32529 

.94561 

.34175 

.93979 

I 

00 

.27564 

.96126 

.29237 

.95630 

.3,0902  .95106 

.32557 

.94552 

34202 

.93969 

0 

M. 

Corfu. 

Sine. 

Cosin. 

Sine. 

Cosin.  '  Sine. 

Coain. 

Sine. 

Cosin. 

Sine. 

M. 

74° 

730       73° 

710 

700 

280 


TAULK 


NATURAL     SINKS     AND    COSINES- 


300 

310 

330 

330 

M 

Bine. 

Coein 

Sine. 

Coein 

Sine.  jCoein 

8in7~ 

Coein 

Sine 

Coein  M 

~0 

34202 

.93989 

358:37 

.933  .IS 

37461 

927|x 

.39073 

92050 

4n674 

.91366  60 

1 

34229 

93D5'J 

.35*61 

.9334- 

.37488 

.92707 

.39100 

.92039 

4o7i  KJ 

91343;  59 

2 

34257 

9394  '.» 

37515 

.92697 

.39127 

9202- 

4f  i7*^7 

91331  1  58 

3 

342-<4 

93931* 

35918 

.93:i27 

.37648 

.926-<«; 

311  153 

92016 

40753 

91319 

67 

4 

34311 

.93929 

3.-.9I5 

.9a3i<> 

.37569 

.92675 

3'M*o 

.92005 

}(l7-0 

.913JC 

56 

5 

343-39 

93919 

a~>973 

.9331  Ri 

.37595 

.92664 

!39207 

91994 

.40*06 

91-295 

56 

6 

34366 

.93909 

36000 

.93-295 

.37622 

.92653 

.39234 

40*33 

912*3!  54 

7 

34393 

.93899 

36027 

.37649 

19*642 

.39260 

91971 

40*60 

91272 

53 

8 

34421 

.93**9 

36054 

!  93274 

.37676 

.92631 

3-J2S7 

91959 

4o**6 

91260 

52 

9 

34148 

93879 

36081 

.93264 

37703 

.92620 

.39314 

9194- 

40913 

91248  51 

10 

34475 

93*6:* 

.36108 

.93253 

.3773d 

.92609 

.39341 

919.% 

40939 

91236150 

11 

34.503 

93859 

.36135 

.93243 

.37757 

.9259* 

.39367 

.91925 

40966 

91-224 

49 

12 

34630 

36162 

.9323-2 

.37784 

.92587 

.39394 

91914 

40992 

.91212 

4* 

13 

34557 

!  93*39 

36190 

.93222 

37811 

.92576 

.39421 

9l9(r2 

41019 

91-200 

47 

14 

34584 

.93>29 

36217 

.93211 

.37838 

.92565 

3944* 

91891 

41045 

46 

15 

.34612 

.93319 

.36244 

.93201 

.37865 

.92554 

.39474 

.91879 

41072 

.91176 

45 

16 

34639 

93-09 

36271 

.93190 

.37892 

.92543 

.39501 

.9136* 

41098 

.91164 

44 

17 

31666 

93799 

36298 

.93180 

37919 

.92532 

.39528 

.91966 

41125 

.91152 

43 

18 

34694 

937S9 

.36325 

.93169 

.37946 

.92.V21 

39555 

91845 

41151 

.91140 

42 

19 

34721 

.93779 

36352 

.93159 

.37973 

.92510 

39581 

91*33 

41178 

.91  1-28 

41 

20 

.34748 

.93769 

.36379 

.93148 

.37999 

.92499 

.39608 

91822 

412(4 

.91116 

40 

21 

34775 

.93759 

.364% 

.93137 

.3S026 

.9248.-! 

.39635 

.91810 

41231 

.91  KM 

39 

22 

34803 

.9374* 

36434 

.93127 

.33053 

.92477 

.39661 

91799 

41  £57 

.91092 

3s 

23 

34830 

.93738 

.36461 

.93116 

.38080 

.92466 

.39688 

.91787 

.412*4 

.910SO 

37 

24 

34857 

.93728 

.364S8 

.93106 

.33107 

92455 

.39715 

91776 

.41310 

9  1  o6» 

36 

25 

34884 

.93718 

.36515 

.93095 

.3^134 

92444 

.39741 

91764 

.41337 

.91056 

35 

26 

34912 

.93703 

.36648 

.930s4 

.38161 

.92432 

39768 

.91752 

.41363 

9KM4 

34 

27 

31939 

.93698 

365(59 

.93074 

.38188 

.92421 

.39795 

.91741 

.41390 

91032 

33 

28 

.34966 

.9368* 

3651H5 

.93063 

.3S2I5 

.92410 

.39*22 

.91729 

.41416 

.  9  1  02;  i 

32 

29 

.34993 

'  93677 

.36623 

.93052 

.33241 

.92399 

.39848 

9171* 

41443 

JWOflH 

31 

30 

.35021 

.93667 

.36669 

.93042 

.3S-26* 

.9238.8 

.39876 

!  91  706 

.41469 

.90996 

30 

31 

.aws 

.93657 

36677 

.93031 

.3*295 

.92377 

.3990T. 

91694 

41496 

909*4 

29 

32 

.35075 

.93617 

,36704 

.93020 

.38322 

.92366 

.39988 

.916*3 

41522 

.90972 

2* 

33 

.35102 

.9:5637 

.36731 

.93010 

.38349 

.923;", 

.39955 

.91671 

41549 

.90960 

27 

34 

.35130 

.93626 

36758 

.92999 

.3*376 

.92343 

.39982 

.91660 

.41575 

.90948 

26 

35 

a5  157 

.93616 

.36786 

.9298S 

.3*403 

.92332 

.40008 

.9164* 

.41*08 

.90936 

25 

36 

a>l84 

.93606 

.36812 

.92978 

.38430 

.92321 

.40035 

.91636 

.41628 

.90924 

24 

37 

!av2ii 

.93596 

.36839 

.92967 

.92310 

.40062 

.91625 

41655 

.90911 

23 

38 

.35239 

.9:3585 

.36367 

.92956 

!Ws3 

92299 

.40088 

.91613 

41641 

.90*99 

22 

39 

aV266 

.93575 

.36S94 

.92915 

.3*510 

.922*7 

4"l  15 

.91601 

.41707 

90**7 

21 

40 

.35293 

.93565 

.36921 

.92935 

38537 

92276 

'40141 

.91590 

.41734 

90876 

2ft 

41 

.35320 

.93555 

369  H 

.92924 

.38564 

.92265 

.4016* 

.91578 

41760 

90863 

19 

42 

.35347 

.93544 

.36975 

.92913 

.38591 

.92254 

.40195 

.91566 

.41787 

.90851 

IS 

43 

.35375 

.93531 

37002 

.92902 

.38617 

.92213 

.40221 

.91555 

.41813 

.90839 

17 

44 

.a5402 

.9*524 

.37029 

.92S92 

.38644 

.92231 

.40248 

.9?  543 

.41840 

.90826   16 

45 

.35429 

.93514 

.37056 

.92881 

.38671 

.92220 

.40875 

.91531 

.41866 

.908141  15 

46 

35456 

.93503 

37083 

.92^70 

3*69* 

.92209 

.40301 

.91519 

41892 

90802 

14 

47 

.35484 

.93493 

37110 

.92859 

.38785 

.9219S 

.40328 

,9l5os 

41919 

90790 

13 

48 

.35511 

.934  S3 

37137 

.92849 

38752 

.921*6 

.40355 

.91496 

.41945 

.90778 

12 

49 

.35538 

.93472 

37164 

.92838 

18778 

.92175 

4i  1331 

.914.84 

41972 

.90766 

11 

50 

.35565 

.9.3462 

37191 

.92S27 

.38305 

.92164 

.40408 

.9147-2 

41998 

.90753 

10 

51 

35592 

.93452 

.37218 

.92816 

.39833 

.92152 

.40434 

91461 

48034 

.90741 

9 

52 

.35619 

.93411 

37245 

.92.S05 

.3*859 

.92141 

.40461 

91449 

42061 

.90729 

8 

53 

.3">647 

.93431 

37272 

.92794 

.38886 

.92130 

.404.88 

91437 

42077 

.90717 

7 

54 

.35674 

.93420 

.37299 

.927S4 

.3*912 

92119 

.40514 

91425 

48104 

907(M 

6 

55 

.35701 

93410 

37326 

.92773 

.38939 

.92107 

.40541 

91414 

42130 

.90692 

5 

56 

.35728 

.93400 

.37353 

.92762 

.38966 

.92096 

40567 

9140* 

42156 

906*0 

4 

57 

.35755 

933*9 

.37380 

.92751 

.3*993 

.92085 

.40594 

.91390 

42  183 

9;  tf.rtf 

3 

63 

35782 

.93379 

.37407 

.92740 

92073 

.40621 

.9137* 

42809 

90655 

o 

59 

35*10 

9336* 

37434 

.92729 

.'  391*46 

.9-2062 

.40647 

91366 

.42836 

.9064.3 

1 

60 

35S37 

.37461 

.92718 

,39073 

.92050 

.40674 

91355 

42*62 

.9(1631 

0 

if 

Coein. 

Sine. 

Oosin. 

Sine. 

Codn. 

Stair 

Coflin. 

Sine. 

Coein. 

Slue. 

M 

690 

680 

670 

660 

65- 

TABLE    III.       XATl'RAI.     SIXES    AXI)    COSIXKS. 


353 

36" 

87° 

28° 

29° 

M. 

Sine. 

CoBln. 

Sine. 

Cosin. 

Blue.    Cosin. 

Sine. 

Cosin. 

Sine. 

Ocein 

M. 

0 

.42262 

.90631 

43337 

89379 

.45399 

.89HU 

.46947 

.88295 

.48481 

.87462 

60 

1 

.422-W 

.9(1618 

.43363 

.89367 

.45425 

.89087 

46973 

.88281 

.48506 

.87448 

59 

2 

.4231  5 

.90*16 

43889 

.89854 

.46451 

.89074 

.46999 

.S8267 

.48532 

.87434 

58 

3 

.42341 

.90194 

.43916 

.89841 

.45477 

.89061 

.47(124 

.88254 

.48557 

.87420 

57 

1 

42:167 

.90.-)32 

.43942 

.89323 

.455(13 

.891*48 

.47050 

.8824'! 

.48583 

.87406 

56 

5 

42394 

90569 

.4:1963 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608 

.87391 

55 

6 

44420 

.90557 

.43994 

.89803 

46664 

.89021 

.47101 

.88213 

.48634 

S7377 

54 

7 

42446 

.9054:. 

44020 

.89790 

45580 

.89008 

.47127 

.88199 

.48659 

.87363 

53 

9 

42473 

.  9(632 

.44046 

.89777 

.15606 

.88995 

.47153 

.88185 

.48684 

87349 

52 

9 

42499 

.90520 

44072 

.89764 

.45632 

.88981 

.47178 

.88172 

.48710 

.87335 

51 

10 

.42525 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

.42552 

.9049o 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

42573 

.90433 

44151 

.89726 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88928 

.47231 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90458 

442f»3 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

44255 

.89674 

.45313 

.88888 

.47358 

.88075 

.48883 

.87235 

44 

17 

.42709 

.90421 

44281 

.8966-2 

.45,839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90408 

44307 

.89619 

.45865 

.88862 

.47409 

.88048 

.48933 

.87207 

42 

19 

42762 

.90396 

44333 

.89636 

.45891 

.88848 

47434 

.88034 

.48964 

.87193 

41 

20 

.42738 

.90333 

.44359 

.89623 

.45917 

.88835 

.47460 

.88(120 

.48989 

.87178 

40 

21 

.42*15 

.90371 

44335 

89610 

.45942 

.88822 

.47486 

.88006 

.49014 

.87164 

39 

22 

.42841 

.90358 

.44411 

.89597 

.45963 

.88808 

.47511 

.87993 

.49040 

.87150 

38 

23 

.42367 

90346 

44437 

.895.84 

45994 

.88795 

.47537 

.87979 

.49065 

.87136 

37 

24 

.42-894 

.9(1334 

44464 

.89571 

.4602') 

.88782 

.47562 

.87965 

.49090 

.87121 

36 

25 

.42920 

.90321 

44490 

.89553 

46046 

.88763 

.47538 

.87951 

.49116 

.87107 

35 

26 

.42948 

.90309 

44516 

.89515 

.46072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90290 

44542 

.895:12 

46097 

.S3741 

.47639 

.87923 

49166 

.87079 

33 

23 

.42999 

.902*4 

44563 

.89519 

.46123 

.88728 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.90271 

44594 

.89506 

.46149 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

.43077 

.90246 

44646 

.89480 

.46201 

.88688 

.47741 

.87863 

.49268 

.87021 

29 

32 

43104 

.9(»233 

41672 

.89467 

.46226 

.&867J 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130  .90221 

44693 

89454 

.46252 

.88661 

.47793 

.87840 

49318 

.86993 

27 

34 

43156  j.  90203 

44724 

.89441 

.46278 

.88647 

.47818 

.87826 

.49344 

.86978 

26 

35 

.431  32  .  90196 

44750 

.89428 

46304 

.836:14 

.47844 

.87812 

49369 

.86964 

25 

36 

.4:1209 

.901  S3 

44776 

.89415 

46330 

.88620 

.47369 

.37798 

491194 

.86949 

24 

37 

.4-1236 

.90171 

44802 

.89402 

46355 

.88607 

.47895 

.87784 

.49419 

.86935 

23 

3* 

.43261 

.90158 

44828 

893*9 

46:i*i 

.88593 

.47920 

.8777(1 

49445 

.86921 

22 

39 

.4.')287 

.90146 

44854 

,89376 

46-407 

.88580 

.47946 

.87756 

49470 

.86906 

21 

40 

.43313 

.90I33 

44880 

.89363 

46433 

.88566 

.47971 

.87743 

49-495 

.86S92 

20 

41 

.43340 

.90120 

.449(t6 

.893.V) 

46453 

.88553 

47997 

.87729 

.49521 

.86878 

19 

42 

.4.3366 

.901118 

44932 

.89337 

46484 

.88539 

.480-22 

.87715 

49546 

.86863 

18 

43 

.43392 

.90095 

.44953 

.89324 

46510 

.88526 

.48043 

.87701 

49571 

.86349 

17 

44 

.43-J13 

.90082 

44984 

.8931  1 

46536 

.88512 

.48073 

.87687 

49596 

.86834 

16 

45 

.43445 

.90071) 

.45010 

.89293 

.46561 

.88499 

.48099 

.87673 

.49622 

.86820 

15 

46 

.43471 

.90057 

45036 

.89285 

.46587 

.8.8435 

.48124 

.87659 

.49647 

.86805 

14 

47 

.43497 

.90045 

45(162 

.89272 

.46613 

.88472 

.48150 

.87645 

.49672 

.86791 

13 

43 

.43523 

.90032 

.46088 

.892*9 

.4C.639 

.88458 

.48175 

.87631 

49697 

.86777 

12 

49 

43549 

.90019 

45114 

.89245 

.4f/,fi4 

.88445 

.48201 

.87617 

49723 

.86762 

11 

50 

43575 

90007 

45HO 

.89232 

.46690 

.88431 

48226 

.87603 

49748 

.86748 

JO 

61 

43602 

.89994 

15166 

.89219 

.46716 

.88417 

.48252 

.87539 

49773 

.86733 

9 

52 

.43628 

89981 

45192 

.89206 

.46742 

.884m 

.48277 

87575 

49798 

.86719 

8 

53 

43654 

.8996* 

4/V2I3 

.89193 

.46767 

.88.390 

.48:103 

.87561 

49824 

.867(»4 

7 

54 

.43680 

.  89956 

45243 

.89180 

.46793 

.88377 

.48328 

.87546 

49*49 

86690 

6 

55 

43706 

.8994.'! 

45269 

.89167 

,46>H9 

.88363 

.483*4 

.87532 

49*74 

.86675 

5 

56 

43733 

.89930 

45295  .89153 

46844 

.88349 

.43379 

.87518 

4<H99 

86661 

4 

57 

43759 

.899H 

.45321 

.89140 

46870 

.88336 

.48405 

.875)14 

.49924 

.86646 

Q 

53 

43785 

.89905 

45.147 

.89127 

.40*6 

.883-22 

.4*430 

.87490 

.49950 

86632 

A 

59 

43SH 

.89882 

.45373 

.89114 

.46921 

.88308 

.48456 

.87476 

49975 

.86617 

1 

60 

43837 

.89879 

.45399 

.8510! 

46947 

.88295 

.43481 

.87462 

.5HOOO 

.86603 

0 

ML 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Coein. 

•SET 

Cosin. 

Sine. 

M 

64o      1 

630 

630 

dio 

600 

TABLE    III.        NATURAL     SINKS     AND    COSINES. 


300 

31<> 

aao 

330 

840 

M. 

Sine. 

CotiD 

Si.i* 

Cosin. 

Sine    jCorin. 

Sine. 

Coeln. 

8ine. 

Gotdn 

M. 

0 

50000 

86603 

51504 

.85717 

52992,.  84805 

.54464 

."83867 

55919 

.*2904 

60 

1 

.50J25 

H65SN 

51629 

.85702 

5»  it  7  1.84  789 

.544^s 

.8.3.S51 

55943 

82N87 

69 

2 

.50)50 

.86573 

51554 

.8MH7 

63041|.84774 

.54513 

.83835 

55968 

.*£87l 

58 

3 

.50076 

86559 

.51679 

.  85*72 

53066 

.o475y 

.64537 

.»;i-5iy 

55992 

.82,855 

57 

4 

50101 

86544 

61004 

.85057 

wwi 

.84743 

.64561 

.H3H« 

56016 

.82*39 

56 

5 

.50126 

8653  1 

Mfl2H 

.3664  X! 

.63115 

.84725 

.:-4.>  o 

.837<e 

56040 

82822 

56 

ft 

.50151 

86515 

51663 

.86«27 

.63140 

.84712 

.64610 

..S3772 

56064 

.82M»6 

54 

7 

.50176 

.86501 

51678 

.86012 

.531  4 

.SMVJ7 

.64635 

.83756 

560SK 

.82790 

53 

g 

.50201 

.864*6 

:»|703 

..H5C97 

631*9 

.844^1 

.54659 

.83740 

.66112 

82773 

52 

9 

.50227 

86471 

.517*3 

.856*2 

63214 

.M666 

.546>3 

.83724 

.56136 

.82767 

51 

10 

.50252 

.86457 

81753 

.a'wfi? 

53.  -38 

.846^J 

.54708 

.837(K 

.56160 

.82741 

60 

11 

50277 

86442 

SITTb 

.86651 

53263 

.S46:«i 

.54732 

.83692 

.561*4 

.82724 

49 

12 

.50302 

86127 

51803 

.85536 

.53288 

.84619 

.54756 

.83676 

.56208 

.S2708 

48 

13 

.50327 

86113 

51828 

.85521 

.53312 

.816.4 

.54781 

.83661' 

.56232 

.82692 

47 

14 

.50352 

86398 

51852 

.8650.; 

.63337 

.8Lrr^S 

.64S06 

.83645 

.66256 

82675 

46 

15 

.60377 

.86384 

.5187; 

.86491 

.53.361 

.84573 

.64829 

.83629 

.66280 

.82659 

45 

16 

.60403 

.86369 

.51902 

.8.',  4  76 

.53386 

.845o7 

.64854 

.83613 

.56305 

.82643 

44 

17 

.6042* 

86354 

.61927 

.85461 

.6.3411 

.84542 

.54^78 

.83597 

.56329 

.82626 

43 

13 

.50453 

.86340 

.61952 

.854  46 

.5:3435 

.84526 

.54902 

.835;!  1 

56353 

.82610 

42 

19 

.5(1478 

.86326 

,61977 

.8-Vj.il 

.53460 

.84511 

.54927 

.83565 

66377 

.82593 

41 

20 

.50503 

.86310 

.62002 

.854  16 

.534*1 

.84495 

.54951 

.83549 

66401 

.82677 

40 

21 

.50528 

.862:* 

.52026 

.awi 

.eaw 

.844  SO 

.54975 

.83533 

56425 

82561 

39 

22 

.50553 

.862*1 

.52'  151 

.85.V5 

.5::).u 

.84464 

.64999 

.83517 

.56449 

.82544 

38 

23 

.50578 

.86266 

.62076 

.85370 

.5355,8 

.844  H 

.55024 

.83501 

56473 

.82528 

37 

24 

.50603 

86251 

.6*2101 

.85355 

.83683 

.84433 

.55048 

.834  >5 

.56497 

.62511 

36 

25 

50628 

86237 

.62126 

.85.40 

.5:5607 

.84417 

.55072 

.83469 

56521 

.82495 

35 

26 

50654 

.Wf££ 

.62151 

.ar>325 

.53632 

.84402 

.55097 

.83453 

.56545 

.8247S 

34 

87 

50679 

88207 

.62175 

.85310 

.53656 

.P-13^6 

.55121 

.83437 

56569 

.82462 

33 

2* 

.60704 

86192 

.62200 

.9521*4 

.63681 

.84370 

.5514.') 

.83421 

56593 

.82446 

32 

29 

60729 

8«178 

.  52225 

.85279 

.53705 

.84355 

.5.r)l6'J 

.83405 

56617 

.82429 

31 

30 

.60754 

.86163 

52250 

.85264 

.63730 

.84339 

.55194 

.83389 

.56641 

.82413 

30 

31 

.50779 

8614- 

52275 

.86249 

,83754 

.84324 

.55218 

.83373 

56665 

.82396 

29 

32 

.50804 

.86111 

62299 

.*:.2;n 

.53779 

.843  H 

.55212 

.83356 

666*9 

.82380 

28 

33 

.50829 

.86119 

52324 

.852  IS 

.68804 

.84292 

.55266 

.83340 

66713 

.82363 

27 

34 

.50854 

.86101 

52349 

.a<5203 

.53838 

.84277 

.55291 

.83324 

56736 

.82:347 

26 

35 

.50K79 

.860*9 

T.2371 

.85188 

.r,3«53 

.84261 

.55315 

.83308 

56760 

.82330 

26 

36 

.60904 

.86074 

52399 

.85173 

.53*77 

.84215 

55339 

.83292 

567S4 

.82314 

24 

37 

50929  ;.  86069 

52423 

.85157 

.53902 

.84^50 

.55363 

.83276 

56^08 

.82297 

23 

3S 

.60954 

86U45 

88449 

.85142 

.53926 

.84214 

.55588 

.83260 

.56832 

.82281 

22 

39 

.50979 

8fl030 

52473 

.85127 

.53951 

.84193 

.55412 

.83244 

56856 

.82264 

21 

40 

61004 

.860(5 

->4>49* 

.85112 

.53975 

.84182 

.66436 

.83228 

5#«0 

.82248 

20 

41 

.61029 

.96000 

52522 

.850% 

.54000 

.84167 

55460 

83212 

569f»4 

.82231 

19 

42 

51054 

859Xft 

52547 

.86081 

.54024 

.84151 

554  s4 

.a?  195 

.56928 

.82214 

18 

43 

.61079 

.8597(1 

52572 

.85066 

.54049 

.84135 

55509 

.83179 

.56952 

.82198 

17 

44 

.61104 

.85956 

52597 

.850ol 

.54073 

.84120 

55533 

.83  W3 

56976 

.82181 

16 

45 

.51129 

.85941 

52621 

.85035 

.54097 

.84104 

.55557 

.83147 

.67000 

.82165 

'6 

46 

61154 

.85926 

52646 

.85020 

.54122 

.84088 

,650)1 

.83131 

.67024 

.82148 

14 

47 

.51179 

.85911 

52671 

.85005 

.54146 

.84072 

.55605 

.831  15 

67047 

.82132 

13 

4* 

.i!204 

.85*96 

52696 

.849*9 

.64171 

.84057 

65S30 

8309« 

57071 

.82115 

r? 

49 

.51229 

85RS  | 

52720 

.84974 

54195 

.84041 

•55654 

.83082 

.67095 

.82008 

11 

60 

51254 

83«e« 

52745 

.84959 

.54220 

.84025 

.6567* 

.83066 

57119 

.89089 

10 

51 

.51279 

85H51 

52770 

.84943 

54244 

.84009 

55702 

.83050 

67143 

.824165 

9 

52 

.61304 

a^3« 

52794 

.8492* 

54269 

.83994 

55726 

.83034 

.67167 

.82048 

8 

53 

51329 

85^21 

MS  19 

.84913 

54293 

.8397* 

65750 

.83017 

.57191 

.82032 

54 

51354 

85soR 

52*44 

.84*97 

54317 

.83962 

55775 

.83001 

.57215 

.82015 

55 

51379 

85792 

52*69 

.8488V! 

.-•4342 

.R3946 

55799 

82986 

.67238 

.81999 

5 

56 

514^ 

.8577? 

52*93 

.84*66 

.W66 

.8393*1 

65823 

82969 

.57262 

8I9R3 

4 

57 

51429 

.857H2 

52918 

84*51 

.54391 

.83915 

55S47 

82953 

57286 

91966 

3 

58 

51454 

.85747 

52943 

.84S3fi 

54415 

.83*99 

.55871 

.829:36 

67310 

81949 

« 

59 

51479 

.85732 

52967 

.84*20 

54440 

.83K.S3 

.56896 

82920 

57334 

.81932 

1 

60 

51504 

.85717 

32992 

.84805 

54464 

.83^57 

55919 

82904 

57358 

81915 

0 

M: 

Oodn. 

Blno. 

Cosin. 

Sice. 

Cosin. 

Sine. 

Coflin. 

Bine 

Cosin. 

Sine. 

M. 

590 

583 

67° 

56° 

550 

T  A  I;].K    Jil.        X.VIt'K.M. 


AXD    COSINES. 


283 


aao 

360 

370 

380 

300 

M. 

Slue. 

Oosln 

Bine. 

Corfu. 

Sine. 

Cosin 

Sine. 

Corin 

Sine. 

Cceiu 

M. 

0 

67358 

.81915 

58779 

.80902 

.60182 

.79*64 

.61566 

178801 

~62932 

TTTlfi 

60 

573*1 

.81899 

.68809 

.80**f> 

60205 

.79846 

.615*9 

.787*3 

62955 

.77696 

69 

'4 

57405 

.81882 

.58826 

.80867 

.60228 

.79S29 

.61612 

.78765 

62977 

.77676 

68 

3 

57429 

.81865 

.58*49 

.80850 

602'>  1 

.79*11 

.61635 

.78747 

63000 

.77660 

67 

4 

57453 

.81848 

.58873 

.80833 

60274 

.79793 

.61658 

.78729 

63022 

.77641 

66 

5 

67477 

.81832 

.68896 

.80*16 

.60298 

.7977C 

.61681 

.78711 

6*45 

.77623 

65 

6 

.57501 

.81815 

.68921 

.80799 

60321 

.79758 

61704 

.78694 

63068 

r,  eos 

64 

7 

.57524 

.81798 

5*943 

.807M2 

6<t:U4 

.79741 

.61726 

78676 

6309( 

.77686 

63 

8 

.57548 

.81782 

5S967 

S0765 

60367 

.79723 

61749 

.78658 

63113 

.77568 

52 

9 

57572 

.81765 

.5*99< 

.8074-i 

6039< 

.79706 

.61772 

.7864( 

63135 

7755(1 

61 

10 

57596 

.8174* 

59014 

.80730 

6IHI4 

.79688 

61795 

.78622 

63I5S 

77531 

60 

11 

.57619 

81731 

59037 

.80713 

60437 

.79671 

61818 

.786(4 

63180 

.77513 

49 

12 

57643!  81714 

59061 

.80696 

,6<H60 

.79653 

.61841 

.78586 

63203 

.77494 

48 

13 

.57667 

.6169* 

.59084 

S067-J 

.6(483 

.79635 

.61*64 

.78568 

63225 

.77476 

47 

14 

67691 

.81681 

59108 

80662 

.60506 

.7961* 

.61887 

.7855( 

.63248 

.77458 

46 

16 

57715 

.81664 

.59131 

.80644 

.60529 

.79600 

.61909 

.78532 

63271 

.77439 

45 

16 

57738 

81647 

59154 

.80627 

60553 

.79583 

.61932 

.78514 

.63293 

.77421 

44 

17 

57762 

.81631 

59178 

.80610 

.60576 

.79565 

.61955 

.78496 

63316 

.77402 

43 

18 

.67786 

.81614 

5920  1 

.80593 

.60599 

.795-17 

.61978 

.78478 

63338 

.77a*l 

42 

19 

57810 

.81597 

59225 

.80576 

.60622 

.79530 

62001 

.78460 

63361 

.77366 

41 

20 

67833 

.81580 

59248 

.8055* 

.60645 

.79512 

.62024 

78442 

63383 

.77347 

40 

21 

67857 

.81563 

59272 

8W541 

.60668 

.79494 

62046 

.78424 

63406 

.77329 

39 

82 

57881 

.81546 

59295 

.80524 

.60691 

.79477 

62069 

.78405 

63428 

.77310 

38 

23 

57904 

.8153H 

59318 

.80507 

.60714 

.79459 

.62092 

.78387 

63451 

.77292 

37 

24 

57928 

.81513 

59342 

.804*9 

.60738 

.79441 

.62115 

.78369 

63473 

.77273 

38 

26 

57952 

.81496 

59365 

.80472 

.60761 

.79424 

.62138 

.78351 

634G6 

.77266 

36 

26 

57976 

81479 

.69389 

80455 

.60784 

.79406 

.621  60 

78333 

63518 

.77236 

34 

27 

57999 

.81462 

59412 

8043S 

.60807 

.793*8 

.62183 

78315 

63640 

.77218 

33 

28 

68023 

.81445 

59436 

80420 

.60830 

79371 

.62206 

78297 

63563 

.77199 

32 

29 

68047 

.8142* 

59459 

.80403 

.60853 

79353 

.62229 

.78279 

63586 

.77181 

31 

3D 

.68070 

.81412 

59482 

.80386 

.60876 

79335 

.62261 

.78261 

63608 

.77162 

30 

31 

58094 

81395 

59506 

.8036S 

.60S99 

79318 

.62274 

.78243 

63630 

.77144 

29 

32 

58118 

81378 

59529 

.80351 

.60922 

79300 

.62297 

.78225 

fi,W>3 

.77125 

2H 

33 

58141 

81361 

59552 

30334 

.60945 

792*2 

.62320 

.78206 

63675 

.771071  27 

34 

58165 

81344 

59576 

80316 

6(1968 

79264 

.62312 

.78188 

63698 

.77088  26 

35 

58189 

81327 

59599 

80299 

.60991 

79247 

.62365 

.78170 

63720 

.770701  25 

36 

58212 

81310 

59622 

802*2 

.61015 

79229 

.62388 

.78152 

63742 

77051  24 

37 

58236 

.81293 

59646 

80264 

61038 

79211 

.62411 

.78134 

.63765 

77033i  23 

38 

58260 

.81276 

59669 

80247 

.61061 

79193 

.62433 

.78116 

63787 

77014  22 

39 

58283 

.81250 

59693 

80230 

61084 

79176 

.62456 

7809« 

.63810 

76996  21 

40 

.58307 

.81242 

59716 

80212 

61  107 

79158 

62479 

78079 

63*32 

76977 

20 

41 

.58330 

.81225 

59739 

80195 

61130 

79140 

62502 

78061 

.63*54 

76959 

19 

42 

.58354 

.812(18 

59763 

80178 

61153 

79122 

62524 

78m3 

.63877 

76940 

18 

43 

58378 

.81191 

59786 

80160 

61176 

79105 

62547 

78025 

63899 

76921 

17 

44 

58401 

.81174 

59809 

80143 

61199 

79087 

62570 

78007 

.63922 

76903 

16 

45 

58425 

.81157 

59832 

80125 

61222 

79069 

88592 

77988 

63944 

76884 

16 

46 

58449 

.81140 

59856 

801  OR 

61245 

79051 

62615 

77970 

63966 

76866 

14 

47 

.58472 

.81123 

59879 

80091 

61268 

79033 

62638 

77952 

63989 

76847 

13 

481.58496 

.81106 

69909 

80073 

61291 

79016 

62660 

77934 

64011 

76828 

12 

49  .58519 

.810*9 

59926 

S0056 

61314 

78998 

626*3 

77916 

64033 

76810 

11 

60  .58543 

.81072 

59949 

8003* 

61337 

789*0 

62706 

77897 

64056 

76791 

10 

61  5a%7 

.8H65 

59972 

80021 

61360 

7*962 

62728 

77879 

64078 

76772 

9 

62!  58590 

.8103- 

59995 

80003 

6(383 

78944 

62751 

77861 

64100 

76754 

8 

63  ,58614 

.81021 

60(119 

799*6 

61406 

7*926 

62774 

77*43 

64123 

76735 

7 

M  .60637 

.8|o«tf 

6fH»42 

795)6* 

61429 

7*908 

62796 

77824 

64145 

76717 

6 

55 

58661 

801^7 

60065 

79951 

61451 

78891 

62*19 

77806 

64167 

76698 

6 

56 

5*6*4 

.b0970 

6flQ89 

799:14 

61474 

78873 

62842 

777*8 

64190 

76679 

4 

57 

58708 

.80953 

601  12 

79916 

.61497 

78855 

62864 

77769 

6421? 

76661 

3 

53 

58731 

.80936 

60136 

79*99 

61520 

78837 

62887 

77751 

64234 

76642 

2 

69 

58755 

80919 

60158 

79**  1 

61543 

78819 

62909 

77733 

642.r»6 

76623 

1 

60 

.58779 

.80902 

60182 

79864 

.61566 

78801 

62932 

77715 

64279 

7604 

0 

jjE 

Coedn. 

Sine. 

Coulri. 

Sine. 

Corfu. 

Slue. 

Cosin. 

Sine. 

Cosin. 

Sin*. 

fit 

540 

680    |    6«° 

51° 

600 

284 


TABLE    III,        NATURAL    SIXES    AND    COSINES. 


4O° 

410 

4?o 

43° 

44° 

M. 

Sine. 

Cosin 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Siue.  |  Cosin. 

M. 

~0 

64279 

.76604 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934 

60 

.64301 

.76.M6 

.65648 

.75452 

.66935 

.74295 

.6*^21 

.73116 

6-J487 

.71914 

59 

2 

.643*3 

.76.->67 

6.-.6M 

.75433 

.66956 

.74276 

.6*-242 

.73098 

.69503 

.71*94 

58 

3 

64.346 

.7654* 

.6.-.072 

.75414 

.66978 

.74*56 

.6*204 

73076 

.69529 

.71873 

57 

4 

64368 

.765.30 

.65694 

.75395 

.66999 

.74237 

.6*'2*r 

.73'  156 

6<J549 

.71853 

56 

5 

.64390 

.76511 

.65716 

.75375 

.67021 

.74217 

.6*:30( 

.73036 

09570 

.71833 

55 

6 

64412 

.76492 

65738 

.75386 

.67(43 

.74198 

.6*327 

73016 

69591 

.71813 

54 

7 
8 

.64435 
.64457 

.76473 
.764  .")."> 

65759 
65781 

.75337 
.75318 

.67064 
.670S6 

.74178 
.74159 

.6S349 
.6S37I 

.72996 
.72976 

.69012 
69033 

.7I792J  53 
.71*72  52 

9 

.64479 

.76436 

.65803 

.75299 

.67107 

.74139 

.68:391 

.7*957 

0%:>4 

.71752151 

10 

.64501 

.76417 

.65825 

.75280 

.67129 

.7412' 

.6*412 

.72937 

69676 

.71732 

50 

11 

.64.V24 

.76398 

65847 

.75261 

.67151 

.74100 

.6*4:34 

.72917 

09696 

.71711 

49 

12 

.64546 

.763SH 

65869 

.75241 

.67172 

.74080 

.6*455 

.72897 

.69717 

.71691 

48 

13 

.64568 

.76361 

65891 

.75222 

.67194 

.74061 

.6S476 

.72877 

.69737 

.71671 

47 

14 

.64.-.9U 

.76342 

.65913 

.75203 

67215 

.74041 

.6*497 

.72857 

.69758 

.7165( 

46 

15 

64612 

.76323 

65935 

.75184 

.67237 

.7402-2 

.68518 

.72837 

.69779 

.7163( 

45 

16 

64635 

76304 

65956 

.75165 

67258 

74002 

6.8539 

.72817 

.69800 

.716K 

44 

17 

64657 

.762S6 

65978 

.75146 

672.80 

.739.83 

68561 

.72797 

.69821 

.71590 

43 

13 

.64679 

.76267 

66000 

.75126 

.67301 

.73963 

.68582 

.72777 

.69842 

.715691  42 

19 

64701 

.76248 

66022 

.75107 

.67323 

.73944 

68603 

.72757 

69862 

.715491  41 

20 

64723 

.76229 

.66044 

.75088 

.  67344 

.73924 

68621 

.72737 

.69883 

.71529 

40 

21 

64746 

.76210 

66066 

.75069 

67366 

.739(M 

6,8645 

.72717 

69904 

.71508 

39 

22 

61768 

.76192 

66088 

.75050 

.67387 

.73885 

.68666 

.72097 

69925 

.71483 

38 

23 

64790 

.76173 

66109 

.75030 

.67409 

.73S65 

.68688 

.72077 

09946 

.71468 

37 

24 

64S12 

.76154 

66131 

.75011 

.67430 

.73846 

68709 

.72657 

09966 

.71447 

36 

25 

64*34 

.761,35 

66153 

.74992 

.67452 

.7*826 

6*73f 

.72637 

89987 

.71427 

35 

26 

64S56 

.76116 

60175 

.74973 

.67473 

.73806 

.68751 

.72617 

.70008 

.71407 

34 

27 

64878 

.76097 

66197 

.74953 

.67495 

.73787 

.6S772 

.72597 

70029 

.71386 

33 

28 

64901 

.7607?- 

66218 

.74934 

67516 

•73767 

68793 

.72577 

70049 

.71366 

32 

29 

649-23 

.760.-><» 

00-240 

.74915 

.67538 

.73747 

68814 

.72557 

70O70 

.71345 

31 

30 

64945 

.76041 

66262 

.74896 

67559 

.73728 

68835 

.72537 

.70091 

.71325 

30 

31 

64967 

.76022 

662*4 

.74876 

.67580 

.73708 

6,8857 

.72517 

70112 

.71onJ5 

29 

32 

6-1989 

.76<KI3 

66908 

.74857 

.67602 

.7:36*8 

68878 

.73497 

70132 

.71284 

2H 

33 

65011 

.759*4 

66327 

.74858 

.67623 

.73609 

68899 

.72477 

70153 

.71264 

27 

34 

65033 

.75965 

66349 

.74318 

.67G45 

.73619 

6*920 

.72457 

70174 

.71243 

26 

35 

65065 

.75946 

66371 

.74799 

.67606 

.73629 

6*941 

.72437 

7i«l95 

.71223 

25 

30 

65077 

.75927 

66393 

.74780 

.676.88 

.73610 

6*902 

.72417 

70215 

.71203 

24 

37 

65100 

75908 

66414 

.74760 

.67709 

.73590 

68983 

.72397 

.?rr236 

.71182 

23 

33 

88129 

.75^89 

66436 

.74741 

.67730 

.73570 

69004 

.72377 

70257 

71162 

22 

39 

65144 

.75870 

66458 

.74722 

.67752 

.73551 

69025 

72357 

7.F277 

71141 

21 

40 

65166 

.  75*51 

66480 

.74703 

.67773 

.73531 

69046 

72I337 

70298 

.71121 

20 

41 

6r,  188 

.75*32 

66501 

.74683 

.67795 

.73511 

6906? 

7'2.'*  17 

70319 

71100 

19 

42 

65210 

.75813 

66523 

.71014 

.67SI6 

.73491 

.69088 

.7229? 

70339 

.71080 

18 

43 

65:02 

.75794 

66545 

.71014 

.67S37 

.73472 

.69109 

.72277 

70360 

.71059 

17 

44 

65254 

.75775 

00566 

.74025 

.67859 

.73452 

69130 

.72-257 

.7038  1 

.71039 

16 

45 

65276 

,75756 

6(5588 

.74606 

.67880 

.73432 

69151 

.72236 

.70401 

.71019 

15 

46 

65298 

.75738 

.66610 

.74586 

.67901 

.73413 

.69172 

.72216 

.70422 

.70998 

14 

47 

6*320 

.75719 

60032 

.74567 

.67923 

73393 

.69193 

.721% 

.70443 

.70978 

13 

48 

.65342 

.  75701  • 

OMO.-13 

.74548 

.67944 

73373 

69214 

.72176 

.70463 

.70957 

12| 

49 

65364 

75681) 

.66675 

.74528 

67905 

73353 

.09235 

.72156 

.70434 

.70937 

11 

50 

65386 

75Gfll 

.66697 

.74509 

679S7 

.73333 

6I)'2.">6 

.721:36 

.70505 

.70916 

10 

51 

65408 

75642 

60718 

.744S9 

68008 

7:0  14 

69277 

.72116 

.70525 

.70*96 

9 

52 

.69430 

758*23 

66740 

.74470 

6*029 

73294 

.69-298 

.72(195 

70546 

.70875 

8 

53 

65452 

75ft  M 

66762 

.74451 

68051 

7:}'274 

69319 

.72"75 

70567 

.70855 

7 

54 

65474 

75585 

60783 

.74431 

68(172 

732."v4 

69:  HO 

72055 

70587 

.70S34 

6 

55 

65496 

75506 

66805 

.74412  68093  .73234 

69361 

.  7'2035 

70608 

.70813 

5 

56 

65518 

75547 

66H27 

.74392 

681  15  .73215 

693S2 

72015 

.70628 

70793 

4 

57 

.65540 

75528 

66S48 

.74373 

.68136 

.73195 

69403 

71995 

.70649 

70772 

3 

58 

65562 

75509 

66870 

74353 

.69157 

.73175 

.69424 

71974 

.70670 

.70752 

2 

59 

65584 

75490 

66891 

743:<4 

.68179 

.73155  .69445 

71954 

.70690 

70731 

I 

60 

.65606  .75471 

66913 

74314 

.68200 

,7,'31  35  s  .69466 

719134 

.70711 

70711 

0 

M. 

Cosin.  Sine. 

Cosin. 

SineT 

Cosin. 

Sine. 

Cosin. 

"sin^r 

Cosin. 

Sine. 

M. 

49° 

48° 

47° 

40° 

45° 

TABLE  IV. 
NATURAL   TANGENTS    AND    COTANGENTS. 


285 


286      TABLE     IV.        NATURAL     TANCiEXTS     AXJ)     COTANGENT 


C 

0 

1 

o 

3 

jo 

}0 

M. 

Tiuig. 

Cotang. 

Taug. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

"OOOOO 

Infinite. 

.01746 

67.29110 

.03492 

28.6363 

(15241 

19.0811 

60 

1 

.00029 

343?  75 

.01775 

56.3T.li6 

.03521 

28.3994 

05270 

18.9755 

59 

2 

.00058 

1718.67 

.01804 

55.4415 

.03550 

28.1664 

.05299 

18.8711 

58 

3 

.OOOS7 

1145.<J2 

.01633 

54.5613 

.03579 

27.9372 

05328 

18.7678 

67 

4 

.00116 

859436 

.01862 

63.70*6 

.036(19 

27.7117 

.05357 

18.6656 

56 

6 

.00145 

687549 

Oli91 

52.8821 

0363* 

27  4899 

05387 

185615 

55 

6 

.00175 

672.957 

.01920 

52.0o07 

.03667 

2727lc 

05416 

18.4C.-J5 

54 

7 

.0021(4 

491.11)6 

.01949 

61.3032 

.03696 

27.0566 

05445 

183055 

53 

8 

.00233 

429.718 

.01978 

50.548.' 

.03725 

26.M50 

05474 

18.2677 

52 

9 

.00262 

381.971 

.02007 

49.8157 

.03754 

266367 

05503 

'8.1708 

51 

1C 

.00291 

3J3.774 

.O2.'i36 

49.103'J 

.03783 

26.4316 

05533 

8.0750 

50 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

05562 

17.9802 

49 

12 

.00349 

2S6.478 

.02(195 

47.7395 

.03S42 

26.0307 

05591 

17.8S63 

48 

13 

.00378 

264.441 

.02124 

47.0853 

O3S71 

25.8348 

05620 

17.7934 

47 

14 

.00407 

215.552 

.02153 

46.4489 

.03900 

25  64  IS 

05649 

17.7015 

46 

15 

.00436 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.6106 

45 

16 

.00465 

214.858 

.02211 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

17 

.(XM95 

202219 

.02240 

44.63-^6 

.039-7 

25.079-5 

.05737 

17.4314 

43 

19 

.00524 

190.1M 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17.3432 

42 

16 

.(¥•553 

1SO.M2 

.02298 

43.5031 

.04046 

247185 

.05795 

17.2558 

41 

14) 

.00582 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05S24 

17.1693 

40 

81 

.00611 

163.700 

.02357 

42.4335 

.041(4 

24.3676 

.05*54 

17.0-37 

39 

22 

.00640 

156259 

023-6 

41.9158 

.04133 

24.1957 

05883 

16.9990 

38 

23 

.00669 

149465 

.02415 

41.4106 

.01162 

24.0263 

05912 

16.9150 

37 

24 

.00693 

143.237 

.02444 

40.9174 

.(M191 

23>593 

.05941 

16.8319 

36 

26 

(10727 

137.507 

.02473 

40.4358 

.04220 

236945 

.05970 

1674% 

36 

26 

.(Xf756 

132.219 

.025*12 

39.%:.  5 

.04250 

23.5321 

05999 

16  66*1 

34 

27 

.OU785 

127.321 

.02531 

39.5059 

.04279 

23371^ 

06029 

16.5S74 

33 

28 

00815 

122.774 

.02560 

39.0568 

.04308 

232137 

.06058 

16.5075 

32 

29 

.00844 

118.540 

.02589 

3S.6177 

.(M337 

230577 

.06087 

164283 

31 

30 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.903* 

.06116 

16.3499 

80 

31 

.00002 

110R92 

.02648 

37.76*6 

.04395 

227519 

.06145 

16.2722 

29 

32 

.00931 

107.426 

.02677 

37.3579 

.04424 

22  6020 

.06175 

16.1952 

28 

33 

.00960 

101  171 

.02706 

36.9560 

.04454 

224541 

.06204 

16.1190 

27 

34 

.009*9 

101.107 

.02735 

36.5627 

.(U4S3 

22.3081 

06233 

16.0435 

26 

35 

.01018 

9S.2179 

.02764 

36.1776 

.04512 

22.1640 

(16262 

15.9687 

25 

36 

.01(47 

954*95 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

37 

.01076 

92.90S5 

.(tt*22 

35.4313 

.04570 

21.8813 

.06321 

15>21  1 

23 

38 

.01105 

90.4633 

.02-?5  1 

35.0695 

.04599 

21.7426 

061r>0 

I5.74S3 

22 

39 

.01135 

88.1436 

.02S81 

34.7151 

.04628 

21.61156 

.06379 

15  6762 

21 

40 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21  47(H 

.06408 

156018 

20 

41 

.01193 

83.8435 

.02939 

34.0273 

.046-47 

21.3369 

.06437 

155310 

19 

42 

.01222 

81.8470 

.02968 

33.6935 

.04716 

21  204'J 

.06467 

154638 

18 

43 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

.06496 

153943 

17 

44 

.01280 

7S.1263 

.03026 

33.W52 

M774 

20  9460 

.06525 

153254 

16 

45 

.01309 

76.39(10 

.03056 

32.7303 

.048(3 

20.8188 

.06554 

15  2571 

15 

46 

.01338 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

065R4 

15.1893 

14 

47 

.01367 

73.1390 

.03114 

32.11*1 

.04*62 

20.5691 

.06613 

15.1222 

13 

48 

01396 

71.6151 

.03143 

3I.R205 

.04*91 

20.4465 

.06642 

1  5  ,'£57 

12 

49 

.01425 

70.1533 

03172 

31.62*1 

.04920 

20.3253 

.06671 

14.9K98 

1! 

5( 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

10 

61 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0*72 

.06730 

14.S596 

62 

.01513 

66.1055 

.03259 

306*33 

.ar>007 

19.9702 

06759 

147954 

63 

.01542 

64.85^0 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14  7317 

64 

.01571 

636567 

.03317 

30  1446 

.05066 

197403 

06817 

146686 

65 

.01600 

62.4992 

03346 

298823 

.05095 

19.6273 

06S47 

146059 

66 

.01629 

61.3*29 

01176 

296245 

(15124 

19.5156 

.06*76 

14.5438 

67 

.01658 

60.305* 

(134(15 

29.3711 

.05153 

19.4051 

06905 

144823 

3 

58 

.016-!7 

59.2659 

.03434 

29.1220 

.051*2 

192959 

.06934 

14.4212 

2 

59 

.01716 

58.2612 

(13463 

28.*77I 

.05212 

19  1*79 

06963 

143607 

1 

60 

.01746 

57.290(1 

.03492 

2*.  6363 

.05241 

19.0811 

.065193 

14.3007 

0 

M. 

Cftaug. 

Taug. 

Cotang. 

Taug. 

Co  tang. 

Tang. 

Cotang. 

Tang. 

M. 

8 

90 

8 

go 

8 

70 

1 

160 

TABLE     IV.        NATURAL     TANGENTS     AND 


COTANGENTS.      287 


*o 

50 

6<> 

T°            | 

• 

Ttuig 

Cotang 

Tang. 

Cotaug. 

Tang. 

Cotaug. 

Ttrng. 

Ootan*.   M. 

-$ 

^06993 

14.3IJ07 

.08749 

11.4301 

.10510 

9.51436 

12278 

8.14436 

60 

1 

.07022 

14.2411 

.08778 

11.3919 

.10540 

9.4*761 

.12308 

8.124S1 

60 

2 

.07051 

14.1821 

.08807 

11.3540 

10569 

9.46141 

12338 

8.10536 

68 

3 

07080 

14.1235 

.08837 

11.3163 

.10599 

9.43515 

12367 

8.06600    57 

4 

07110 

14.0655 

.08666 

11.2769 

.10628 

9.40904 

12397 

8.06674    66 

6 

.07139 

14.0079 

.08895 

11.2417 

106.37 

9.36307 

12426 

8.04756 

56 

6 

.07168 

13.9507 

.08925 

ll.»M" 

.10687 

9.35724 

12456 

8.02848 

54 

7 

.07197 

13.8940 

.06954 

11.1681 

.10716 

9.33155 

.12486 

8.00948 

53 

8 

.07227 

13.8378 

.08983 

11.1316 

.10746 

9.30599 

12515 

7.99058 

52 

9 

.07256 

13.7821 

.09013 

11.0954 

.10775 

928056 

12544 

7.97176 

61 

10 

07265 

13.7267 

.06049 

11.0594 

.•10805 

9.25530 

12574 

7.95302 

50 

11 

.07314 

13.67J9 

.09071 

11.0237 

.10834 

9.23016 

12603 

7.93438 

49 

12 

.07344 

13.6174 

.09101 

10.9S.S2 

.10863 

9.20516 

12633 

7.915S2 

48 

13 

.07373 

13.5634 

.09130 

10.9529 

.10893 

9.18026 

12662 

7.89734 

47 

14 

.07402 

13.509^ 

.09159 

10.9178 

.10922 

9.15554 

12692 

7.87895 

46 

15 

.07431 

13.4566 

.09189 

10.8829 

10952 

9.13093 

12722 

7.86064 

45 

16 

07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

12751 

7.84242 

44 

17 

07490 

13.3515 

.09247 

10.8139 

.11011 

9.0821  1 

12781 

7.82428 

43 

18 

.07519 

13.2996 

.09277 

10.7797 

.11  WO 

9.05789 

12810 

7.80622 

42 

19 

.07548 

13.24SO 

093U6 

10.7457 

.11070 

9.03379 

12840 

7.7S825 

41 

20 

.07578 

13.1969 

09335 

10.7119 

.11099 

9.00983 

12669 

7.77036 

40 

21 

07607 

13.1461 

09365 

10.6783 

.11128 

8.98598 

12899 

7.75254 

39 

22 

.07636 

13.0956 

09394 

10.6450 

.11158 

8.96227 

12929 

7.73480 

38 

23 

.07665 

13.0458 

09423 

10.6118 

.11187 

8.93667 

12958 

7.71715 

37 

24 

.07695 

12.9962 

09453 

10.5789 

.11217 

8.91520 

129S3 

7.69957 

36 

25 

.07724 

12.9469 

.09482 

10.5462 

.11246 

8.89185 

13017 

7.68208 

36 

26 

.07753 

12.8981 

09511 

10.6136 

.11276 

8.86362 

13047 

7.66466 

34 

27 

.07782 

12.8496 

09541 

10.4813 

.11305 

8.84551 

13076 

7.64732 

33 

23 

.07812 

12.8014 

09570 

10.4491 

.11335 

8.82252 

13106 

7.630(15 

32 

29 

.07841 

12.7536 

.06600 

10.4172 

.11364 

8.79964 

13136 

7.61287 

31 

30 

.07870 

12.7062 

09629 

10.3854 

.11394 

8.77689 

13165 

7.59576 

30 

31 

.07899 

12.8591 

.09658 

10.3538 

11423 

8.75425 

13195 

7.67872 

29 

32 

.07929 

12.6124 

09688 

10.3224 

.11452 

8.73172 

13224 

766176 

28 

33 

.07258 

12.5660 

.09717 

10.2913 

.11482 

8.70931 

13254 

7.54487 

27 

34 

.07987 

12.5199 

09746 

10.2602 

.11511 

8.68701 

13284 

7.52806 

26 

35 

.08017 

12.4742 

09776 

10.2294 

.11541 

8.664S2 

13313 

7.51132 

26 

36 

.08(146 

12.4288 

09805 

10.1988 

.11570 

8.64275 

13343 

7.49465 

24 

37 

.08075 

12.3838 

09834 

10.1683 

.11600 

8.62078 

13372 

7.47806 

23 

38 

.08104 

12.3390 

.09864 

10.1381 

.11629 

8.59693 

13402 

7.46164 

22 

39 

.08134 

12.2W6 

.09893 

10.1080 

.11659 

8.57718 

13432 

7.44509 

21 

40 

.08163 

12.2505 

09928 

10.0780 

.11688 

8.55555 

13461 

7.42871 

20 

41 

.08192 

12.2067 

09952 

10.0483 

.11718 

8.53402 

13491 

7.41240 

19 

42  |  .08221 

12.1632 

.09981 

10.0187 

.11747 

8.51259 

13521 

7.39616 

18 

43 

.08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

13550 

7.37999 

17 

44 

.08280 

12.0772 

.10040 

9.96007 

11806 

8.47007 

13580 

7.36389 

16 

45 

.08309 

12.0346 

.10069 

9.93101 

.11836 

8.44896 

13609 

7.34786 

16 

46 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

13639 

7.33190 

14 

47 

.08368 

11.9504 

.10128 

9.87338 

.11995 

8.40705 

13669 

7.31600 

13 

48 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.36625 

13698 

7.30018 

12 

49 

.08427 

11.8673 

.10137 

9.81641 

.11954 

8.36555 

13728 

7.26442 

11 

50 

.08456 

11.8262 

.10216 

9.76817 

.11983 

8.34496 

13758 

7.26873 

10 

51 

.08485 

11.7853 

.10246 

9.76009 

.12013 

8.32446 

13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.73217 

.12H42 

8.30406 

13817 

7.23754 

8 

53 

.08544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

13846 

7.22204 

7 

54 

.08573 

116645 

.10334 

9.67680 

.12101 

8.26355 

13876 

7.20661 

6 

81 

.08602 

11.6248 

.10363 

9.64935 

.12131 

8.24345 

139<i6 

7.19125 

6 

56 

08632 

11.5853 

.10393 

9.62205 

12160 

8.22344 

13935 

7.17594 

4 

67 

086fil 

11.5461 

.10422 

9.  594  91  » 

12190 

8.20352 

13965 

^  16071 

3 

58 

0^690 

11.5072 

.10452 

9.56791 

12219 

8.18370 

13995 

M4553 

2 

59 

.08720 

11.4685 

.10481 

9.541(16 

12249 

8.16398 

14024 

7.13042 

1 

60     .08749 

11  4301 

.10510 

9.51436 

.12278 

8.14435 

.14054 

7.  1  1537 

0 

M.Cottuig. 

Tbug. 

Cotmitf. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

s: 

S5° 

84° 

830 

8»o 

TA;;I  F    iv.      NATURAL    TANGENTS    AND    COTAN<;KNT<. 


8° 

1 

JO 

1 

GO 

1 

lio 

If. 

Tang. 

Cotaug. 

Tang. 

Cotang. 

Tang. 

Cotang 

Tang. 

Cotang. 

M. 

0 

14*054 

7.11537 

.15838 

6.31375 

.17633 

5.67128 

19438 

5.14456 

60 

.14084 

7.  UK  ttS 

15868 

6.9M.89 

.17663 

5.66165 

.19468 

5.  13658 

69 

2 

.14113 

7.08546 

.15*98 

6.29007 

.17693 

5.65205 

.19498 

5.12^62 

58 

3 

.14143 

7.07<I59 

.15928 

6.27829 

.17723 

5.C4248 

.19529 

5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

6.63295 

.19559 

5.11279 

56 

5 

.14202 

7.04105 

.]59v88 

6.2.V1-6 

.17783 

5.62344 

19589 

5.  10490 

55 

6. 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.613'J? 

.19619 

5.09701 

54 

7 

.14262 

7.01174 

.16047 

6.2316H 

.17>43 

5.60452 

1<J649 

6.08921 

53 

1 

.1429! 

6.99718 

16077 

6.22H03 

.17873 

6.59511 

li)6>0 

5.08139 

52 

9 

.14321 

T.9826S 

.16107 

6.20S&I 

.17903 

5.&s;,73 

19710 

5.07360 

61 

10 

.14351 

6.96823 

16137 

6.19703 

.179.53 

55763S 

19740 

5.06584 

50 

II 

.14381 

6.  9538;» 

16167 

6.18559 

17'J63 

5.56?<»G 

19770 

5.05809 

49 

./ 

.14410 

6.93952 

16196 

6.17419 

17993 

6.55777 

19801 

5.05037 

48 

13 

.14440 

6.92525 

16226 

6.1G2N3 

.18023 

5.54851 

.19831 

5.04267 

47 

14 

.14470 

6.91104 

16256 

6.15151 

J8063 

5.53927 

19^51 

503499 

46 

16 

14499 

6.is96&> 

16286 

6.14023 

.18083 

5.53007 

19891 

5.02734 

46 

16 

14529 

6.R8278 

16316 

6.12S99 

18113 

6.52090 

19921 

6.01971 

44 

17 

14559 

6.86874 

16346 

6.11771* 

.18143 

6.51176 

19952 

5.01210 

43 

18 

14538 

6.85475 

16376 

6.I06G4 

.18173 

5.50264 

19'J^2 

500451 

42 

19 

.14618 

6.840S2 

16405 

6.09552 

.18203 

5  49356 

2(K)12 

4.99695 

41 

20 

14648 

6.82694 

16435 

6.084-14 

.18233 

5.-4M.';! 

.2rt«2 

4.98940 

40 

91 

14678 

6.81312 

16165 

6.07340 

IS263 

5.4754^ 

20073 

498188 

39 

22 

14707 

6.  79936 

1&495 

6.062411 

18293 

6.4664H 

20103 

497438 

38 

23 

14737 

6.78564 

16525 

6.05143 

.18323 

5.45751 

.20133 

4  96690 

37 

24 

14767 

6.77199 

16555 

6.04:  151 

.18353 

5.44>57 

20164 

495945 

36 

26 

14796 

6.75->38 

16585 

6.02962 

It>3~i4 

5.43-J66 

20194 

4.yfi20| 

35 

2fi 

M-<26 

6.  744  S3 

.16615 

60l8?s 

1K4I4 

6.4»>77 

2ir224 

494460 

34 

27 

14866 

673133 

16645 

6.007-J7 

.18444 

5.42192 

2ir2.-v| 

493721 

33 

28 

14HS6 

6.71789 

16674 

6.99720 

.18474 

6.41309 

2ir/K5 

492984 

32 

29 

14915 

67W50 

16704 

5.9S6-16 

1-fKH 

540429 

20315 

4  92249 

31 

'JO 

14945 

6.69116 

16734 

5.97576 

.18534 

6.39552 

20345 

491516 

30 

31 

14976 

667787 

16764 

596510 

.18564 

53^577 

20376 

4.90785 

29 

32 

15015 

6  66-163 

16794 

5.9544s* 

.1-S5W 

G.  37*05 

2W(« 

4  .90056 

28 

33 

15034 

665144 

16824 

5.94:WO 

.l-«24 

5.36y:« 

.2TM36 

4  893:*) 

27 

34 

15064 

663S3I 

16854 

6.93335 

i^:>4 

5.36070 

2(Mn6 

4.S.X06 

26 

35 

15094 

6.62523 

10884 

6.922S3 

1<6>4 

6.35206 

.2IM97 

4.878W2 

25 

36 

15124 

661219 

16914 

5.9123G 

.18714 

6.34.115 

20527 

4.87162 

24 

37 

15153 

6.59921 

16944 

6.9f»19l 

18745 

5.3:M>7 

20557 

486444 

23 

38 

15183 

6.&S62? 

16974 

5.89151 

.18775 

5.32631 

20.'>88 

485727 

22 

39 

15213 

6.57339 

17004 

6.88114 

.18805 

5.31  77s 

20613 

4.S50I3 

21 

40 

.15243 

6561)55 

17033 

6.8708O 

.18*35 

5.3092S 

.20648 

484300 

20 

41 

15272 

6.54777 

17063 

5.860r>l 

1  <-*65 

530(K) 

20679 

4.83590 

19 

42 

.15302 

6.C35TI3 

17093 

5.85024 

1  -i-yr, 

5.29235 

.20709 

4.82882 

18 

43 

15332 

6.52234 

17123 

5.84(101 

.H92.r> 

5.2.S393 

.20739 

482176 

17 

44 

15362 

6.5W70 

.17153 

6.829^ 

.18955 

5.27553 

.20770 

481471 

16 

46 

.15391 

6.49710 

.17183 

6.81966 

.18986 

6.26715 

.20800 

4.80769 

16 

46 

.15421 

6.4*456 

17213 

6.809r<3 

.190)6 

6.25880 

.20830 

4.80088 

14 

47 

.15451 

6472116 

.17213 

5.79iM4 

1W*16 

6.25(M8 

20861 

4.79370 

13 

48 

154S1 

6.4f>96l 

17273 

5.7^9.'^ 

.19076 

524218 

.20891 

4.7H673 

18 

49 

.15511 

6.44720 

17303 

5.779.16 

19106 

5.23391 

20921 

4.77978 

11 

60 

.15540 

6434^4 

.17333 

6.  76937 

19136 

5.22566 

20952 

4.772«6 

10 

51 

.15570 

642253 

.17363 

5.75U4I 

19106 

5.21744 

20982 

4.76596 

52 

.15600 

6.41026 

17393 

5.74949 

19197 

6  20925 

2KU3 

4.76910 

53 

.156.10 

639804 

17423 

5.73«J60 

.19227 

5.20107 

21043 

475219 

64 

.15660 

6.38M7 

17453 

5.72974 

19257 

6.19293 

.21(173 

474534 

65 

15689 

637374 

17463 

5.71992 

I92H7 

5.IH4SO 

21104 

4.73851 

66 

.15719 

6.36165 

17513 

5.71013 

19317 

6.17671 

21134 

4.73171' 

57 

.15749 

6.349fil 

17543 

6.70037 

19:M7 

5.I&H63 

21164 

4.72490 

3 

58 

.15779 

6.337RI 

17573 

5.69064 

19378 

5  lOfis 

21195 

471813 

2 

69 

.15X19 

6.3*566 

.17603 

5.6S<»94 

19408 

6.15256 

.21225 

4.71137 

1 

60 

.15838 

6.31375 

17633 

5.6712« 

19438 

5.14455 

21256 

470463 

0 

M 

Cotang. 

Tmug. 

Ootang. 

Tang. 

Cotang' 

Tang. 

Ootuug. 

Tang. 

M. 

8 

10 

8 

OQ 

7! 

9ft 

7 

RO 

TABLE     IV.       NATURAL     TANGENTS     AND     COTANGENTS.      289 


I            » 

90 

1 

3° 

1 

*° 

1 

50 

M 

Tang. 

Cotang. 

iTwig. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotung 

M. 

0 

21256 

4.70463 

23087 

4.33148 

7124933 

4.01078 

.26795 

3.73205 

60 

1 

21  486 

4697'JI 

.23117 

4.32573 

.24964 

4.00582 

.26826 

3.72771 

59 

2 

21316 

4.69121 

.23148 

4.321  101 

.24995 

4.D0086 

.26657 

3.72338 

58 

3 

21347 

168452 

23179 

4.31430 

.25026 

3.99592 

.26888 

3.71907 

57 

4 

21377 

467766 

23*19 

4.30S60 

.25056 

3.9909'J 

.26920 

3.71476 

56 

5 

214(18 

4.67121 

.23240 

4.30291 

.25087 

3.93607 

.26951 

3.71046 

65 

6 

21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

54 

7 

21469 

465797 

23301 

4.29159 

25149 

3.97627 

.27013 

3.70188 

53 

8 

21499 

465138 

23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

52 

9 

21529 

464480 

23363 

4.28032 

25211 

3.96651 

.27076 

3.69335 

51 

10 

21560 

463825 

23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

21590 

463171 

23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

12 

21621 

462518 

23455 

4.26352 

.25304 

3.95196 

27169 

3.68061 

48 

13 

21651 

461868 

234  85 

4.25795 

.25335 

3.94713 

27201 

3.67638 

47 

14 

21682 

461219 

23516 

4.25239 

.25366 

3.94232 

.27232 

3.67217 

46 

15 

21712 

4.60572 

23547 

4.24685 

.25397 

3.93751 

.27263 

3.66796 

45 

16 

21743 

459927 

23578 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

44 

17 

21773 

4  592*3 

23608 

4.23580 

.25459 

3.92793 

27326 

3.65957 

43 

18 

21804 

456641 

23639 

4.23030 

.25490 

3.92316 

27357 

3.65538 

42 

19 

21834 

4.58001 

23670 

4.22481 

25521 

3.91839 

27388 

3.65121 

41 

20 

21864 

457363 

23700 

4.21933 

.25552 

3.91364 

27419 

3.64705 

40 

21 

21*95 

4  56726 

23731 

4.21387 

.25583 

3.90890 

27451 

3.64289 

39 

22 

21925 

456091 

23762 

4.20842 

.25614 

3.90417 

27482 

3.63874 

38 

23 

21956 

455458 

23793 

4.20298 

.25645 

3.89945 

27513 

3.63461 

37 

24 

21986 

454826 

23823 

.19756 

.25676 

3.89474 

27545 

3.63048 

36 

25 

22017 

454196 

.23864 

.19215 

.25707 

3.89004 

27576 

3.62636 

36 

26 

22047 

453568 

.23885 

.18675 

.25738 

3.88536 

27607 

3.62224 

34 

27 

.22078 

4.52941 

23916 

.18137 

25769 

3.88068 

27638 

3.61814 

33 

28 

22108 

4.52316 

23946 

.17600 

.25800 

3.87601 

27670 

3.61405 

32 

29 

.22139 

451693 

.23977 

.17064 

25831 

3.87136 

27701 

3.60996 

31 

30 

22169 

4.51071 

24008 

.16530 

25862 

3.86671 

27732 

3.60588 

30 

31 

.22200 

4.50451 

24039 

.15997 

25893 

3.86208 

27764 

3.60181 

29 

32 

22231 

4.49832 

24(169 

.15465 

.25924 

3.85745 

27795 

3.59776 

28 

33 

22261 

4.49215 

24100 

.14934 

.25955 

3.85284 

27826 

3.59370 

27 

34 

22292 

4.48600 

24131 

.14405 

.25986 

3.84824 

27888 

3.58966 

26 

35 

22322 

4.47986 

24162 

.13877 

.26017 

3.84364 

27889 

3.58562 

25 

36 

22353 

4.47374 

24193 

.13350 

26048 

3.83906 

27921 

3.5816U 

24 

37 

22383 

4.46764 

24223 

.12825 

.26079 

3.83449 

27952 

3.57758 

23 

33 

22414 

4.46155 

24254 

.1230! 

.26110 

3.82992 

27983 

3.57357 

22 

39 

22444 

4.45548 

24285 

.11778 

.26141 

3.82537 

28015 

3.56957 

21 

40 

22475 

4  44942 

24316 

.11256 

.26172 

3.82083 

28046 

3.56557 

20 

41 

22505 

4.44338 

24347 

.10736 

.26203 

3.81630 

28077 

3.56159 

19 

42 

22536 

443735 

24377 

.10216 

.26235 

3.81177 

.28109 

3.55761 

18 

43 

22567 

443134 

24408 

4.09699 

.26266 

3.80726 

28140 

3.55364 

17 

44 

22597 

442534 

24439 

4.09182 

.26297 

3.80276 

28172 

3.54968 

16 

45 

22628 

4.41936 

24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

46 

22658 

4.41340 

24501 

4.08152 

.26359 

3.79378 

28234 

3.54179 

14 

47 

226S9 

4.40745 

24532 

4.07639 

26390 

3.78931 

28266 

3.53785 

13 

48 

22719 

4.40152 

24562 

4.07127 

.26421 

3.78485 

.28297 

3.53393 

12 

49 

22750 

439560 

24593 

4.06616 

.26452 

3.78040 

.28329 

3.53001 

11 

50 

22781 

4.38969 

24624 

4.06107 

.26483 

3.77595 

28360 

3.52609 

10 

51 

22811 

4.aS381 

24655 

4.05599 

26515 

3.77152 

28391 

3.52219 

9 

52 

22842 

4.37793 

246^6 

4.05092 

26546 

3.76709 

28423 

3.51829 

8 

53 

22872 

4.37207 

24717 

4  04586 

26577 

3.76268 

.28454 

3.61441 

7 

54 

22903 

436623 

24747 

4.040H1 

26608 

3.75828 

28486 

3.51053 

6 

65 

22934 

4  36(140 

24778 

4.03578 

26639 

3.75388 

28517 

3.50666 

6 

66 

22964 

4»>I59 

24S09 

4.03076 

26670 

3.74950 

28549 

3.50279 

4 

57 

22995 

434879 

24«40 

4.02574 

26701 

3.74512 

.28580 

3.49894 

3 

58 

23W8 

434300 

24871 

402074 

26733 

3.74075 

.28612 

3.49509 

f 

69 

.23056 

433723 

24902 

4.01576 

26764 

378640 

28643 

3.49125 

1 

60 

23H87 

4.33148 

24933 

4.01078 

26795 

3.73205 

.28675 

3.48741 

0 

M. 

Totting. 

Tang. 

Cotang. 

Tang 

Coteng: 

Tang. 

Cotang. 

Tang. 

M. 

1 

T° 

7 

60 

7 

50 

7 

EO 

290      TABLE     IV.        NATURAL     TANGENTS     AND     COTANGENTS. 


i 

GO 

1 

to 

] 

go 

1 

90 

M. 

Ttoig. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

~0 

.28675 

3.48741 

.30573 

3.27085 

.32492 

~3707768 

.34433 

2.90421 

60 

1 

.28706 

3.48359 

.30605 

3.26745 

.32524 

3.07464 

.34-165 

2.90147 

59 

2 

.28738 

3.47977 

.30637 

3.26406 

.32556 

3.07160 

.34498 

2.89873 

58 

3 

.28769 

3.47596 

.30669 

3.26067 

.32588 

3.06857 

.34530 

2.89600 

57 

4 

.28800 

3.47216 

.30700 

3.25729 

.32621 

3.06554 

.34563 

2.89327 

56 

6 

.28832 

3.46S37 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.89055 

55 

6 

.28864 

3.46458 

.30764 

3.25055 

.32685 

3.05950 

.34628 

2.88783 

64 

7 

.28895 

3.46080 

.30796 

3.24719 

.32717 

3.05649 

.34661 

2.8851  1 

53 

8 

.28927 

3.45703 

.30823 

3.24383 

.32749 

3.05349 

.34693 

2.88240 

52 

9 

.28958 

?.45327 

.30860 

3.24049 

.32782 

3.05049 

.34726 

2.87970 

51 

10 

.28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2.87700 

50 

11 

.29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.87430 

49 

12 

.29053 

3.44202 

.30955 

3.23048 

.32878 

3.04152 

.34824 

2.87161 

48 

13 

.29084 

3.43329 

.309S7 

3.22715 

.32911 

3.03854 

.34856 

2.86892 

47 

14 

.29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556 

.34889 

2.86624 

46 

15 

.29147 

&43084 

.31051 

3.22053 

.32975 

3.03260 

.34922 

2.86356 

45 

16 

29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089 

44 

17 

29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.85822 

43 

18 

.29242 

3.41973 

.31147 

3.21063 

.33072 

3.02372 

.35020 

2.85555 

42 

19 

.29274 

3.41601 

.31178 

3.20734 

.33104 

3.02077 

.35052 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.204% 

.33136 

3.01783 

.35085 

2.85023 

40 

21 

.29337 

3.40869 

.31242 

3.20079 

.33169 

3.01489 

.35118 

2.84758 

39 

22 

.29363 

3.40502 

.31274 

3.19752 

.33201 

3.01196 

.35150 

2.84494 

38 

23 

.29400 

3.40136 

.31306 

3.19426 

.33233 

3.00903 

.35183 

2.84229 

37 

24 

.29432 

3.39771 

.31338 

3.19100 

.33266 

3.00611 

.35216 

2.83965 

36 

25 

.29463 

3.39406 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702 

36 

26 

.29495 

339042 

.31402 

3.13451 

.33330 

3.00028 

.35231 

283439 

34 

27 

.29526 

3.38679 

.31434 

318127 

.33363 

2.99738 

.35314 

2.83176 

33 

28 

.29558 

3.38317 

.31466 

3.17804 

.33395 

299447 

.35346 

2.82914 

32 

29 

.29590 

3.37955 

.31498 

3.17481 

.33427 

299158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98668 

.35412 

2.82391 

30 

31 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

32 

.29685 

3.36875 

.31594 

3.16617 

.33521 

2.98292 

.35477 

2.81870 

2S 

33 

.29716 

3.36516 

.31626 

3.16197 

.33557 

2.98004 

.35510 

281610 

27 

34 

.29748 

3.36158 

.31653 

3.15877 

.33589 

2.97717 

.35543 

2.81350 

26 

35 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

35576 

2.81091 

25 

36 

.29811 

335443 

.31722 

3.15240 

.33654 

2.97144 

35608 

2.80833 

24 

37 

.29843 

3.35087 

.31754 

3.14922 

.336S6 

2.96858 

35641 

2.80574 

23 

38 

.29875 

3.34732 

.31786 

3.14605 

.33718 

2.  965~3 

35674 

2.80316 

22 

39 

.29906 

3.34377 

.31818 

3.14288 

.33751 

2.96288 

.35707 

2.80059 

21 

40 

.29933 

3.34023 

.31850 

3.13972 

.33783 

2.960'U 

35740 

2.79802 

20 

41 

.29970 

333670 

.31882 

3.13656 

.33816 

2.95721 

35772 

2.79545 

19 

42 

.30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.35805 

2.79289 

18 

43 

.30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155 

.35838 

2.79033 

17 

44 

.30065 

3.32614 

.31978 

3.12713 

.33913 

2.94872 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400 

.33945 

2.94591 

.35904 

2.78523 

15 

46 

.30128 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47 

.30160 

3.31565 

.32074 

3.11775 

.34010 

2.94028 

.35969 

2.78014 

13 

48 

.30192 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

12  ' 

49 

.30224 

3.30863 

.32139 

3.11153 

.34075 

2.93468 

.36035 

2.77507 

11 

50 

.30255 

3.30521 

.32171 

3.10842 

.34108 

2.93189 

.36068 

2.77254 

10 

51 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

36134 

2.76750 

53 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

54 

.30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

55 

.30414 

3.28795 

.32331 

3.0929S 

.34270 

2.91799 

36232 

2.75996 

56 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

57 

.30478 

3.28109 

.32396 

3.08685 

.34335 

2.91246 

.36298 

2.75496 

58 

.30509 

3.27767 

.32428 

3.08379 

.34363 

2.90971 

.36331 

2,75246 

59 

.30541 

3.27426 

.32460 

3.08073 

.34400 

2.90696 

.36364 

2.74997 

60 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

.36397 

274748 

0 

M: 

Cotaug. 

Tang. 

Cotang. 

Tang. 

Cotang 

Tang. 

Cotuug. 

Tang. 

M. 

V 

30 

7 

»o 

y 

10 

T 

OQ 

TABLE 


NATURAL    TANGENTS    AND    COTANGENTS.      291 


20° 

31° 

aa° 

23° 

M. 

Tang. 

Cotang. 

Tang 

Gotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.36397 

2.74748 

.38386 

2.60509 

.40403 

~!2.47509 

.42447 

2.35585 

60 

1 

.3&430 

2.74499 

38420 

2.60283 

.40436 

2.47302 

.42482 

2.35395 

59 

2 

.36463 

2.74251 

38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

68 

3 

.36496 

2.74004 

3S4*7 

2.59h3l 

.40504 

2.46888 

.42551 

2.35015 

67 

4 

.36529 

2  73756 

38520 

2.59606 

.40538 

2.46682 

.42585 

2.34825 

56 

5 

.36562 

2.73509 

38553 

2.593S1 

.40572 

2.46476 

.42619 

2.34636 

55 

6 

.36595 

2.73*63 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

7 

.36628 

2.7»U7 

38620 

2.58932 

.40640 

2.46065 

42688 

2.34258 

53 

8 

.36661 

2.72771 

38654 

2.58708 

.40674 

2.45860 

.42722 

2.34069 

52 

9 

.36694 

2.72526 

38687 

2.58484 

.40707 

2.45655 

.42757 

2.33881 

51 

10 

.36727 

2.72281 

38721 

2.58261 

.40741 

2.45451 

42791 

233693 

50 

11 

.36760 

2.72036 

38754 

2.58038 

.40775 

2.45246 

42826 

233505 

49 

12 

.36793 

2.71792 

38787 

2.57815 

.40809 

2.45043 

42860 

233317 

48 

13 

36826 

2.71548 

38821 

2.57593 

.40843 

2.44839 

42894 

2.33130 

47 

14 

.36859 

2.71305 

38854 

2.57371 

.40877 

2.44636 

42929 

2.32943 

46 

15 

.36892 

2.71062 

38888 

2.57150 

.40911 

2.44433 

42963 

2.32756 

45 

16 

36925 

2.70819 

38921 

2.56928 

.40945 

2.44230 

42998 

2.32570 

44 

17 

.36958 

2.70577 

38955 

2.56707 

.40979 

2.44027 

43032 

2.3-2383 

43 

18 

36991 

2.70335 

38988 

2.56487 

.41013 

243825 

43f»67 

2.32197 

42 

19 

37024 

2.70094 

39022 

2.56266 

.41047 

2.43623 

43101 

2.32012 

41 

20      37057 

2.69853 

39055 

2.56046 

.41081 

243422 

43136 

2.31826 

40 

21      37090 

269612 

39039 

2.55827 

.41115 

2.43220 

43170 

2.31641 

39 

22      37123 

2.69371 

39122 

2.55608 

.41149 

2.43019 

43205 

2.31456 

38 

23      37157 

2.69131 

39156 

2.553S9 

41183 

2.42819 

43239 

2.31271 

37 

24      37190 

2.68892 

.39190 

2.55170 

41217 

2.42618 

43274 

2.31086 

36 

25!    37223 

2  68653 

39223 

2.  54952 

.41251 

2.42418 

43308 

2.30902 

35 

26      37256 

2.68414 

39257 

2.54734 

.41285 

242218 

43343 

2.30718 

34 

27      37239 

2.68175 

39290 

2.54516 

.41319 

2.42019 

43378 

230534 

33 

28      37322 

2.67937 

39324 

2.54299 

.41353 

2.41819 

43412 

2.30351 

32 

29      37355 

2.6770U 

39357 

2.54082 

.41387 

2.41620 

43447 

2.30167 

31 

30      373S8 

2.67462 

39391 

2.53865 

.41421 

2.41421 

43481 

2.29984 

30 

31      37422 

2.67225 

W25 

2.53648 

.41455 

241223 

43516 

2.29801 

29 

32      37455 

2.66989 

3^458 

2.53432 

.41490 

2.41025 

43550 

2.29619 

28 

33      37488 

2.66752 

3iH^< 

2.53217 

.41524 

2.40827 

43585 

2.29437 

27 

34      37521 

2.66516 

395VK 

2.53001 

.41558 

2.40629 

43620 

2.29254 

26 

35      37554 

2  66-2*1 

39559 

2  52786 

.41592 

2.4043-2 

43654 

2.29073 

25 

36      37588 

2.66(1-46 

39593 

2.52571 

41626 

2.40235 

43689 

2.28891 

24 

37      37621 

2.65811 

39626 

252357 

.41660 

2.40038 

43724 

2.28710 

23 

1  38      37654 

265576 

39660 

2.52142 

41694 

2.39841 

43758 

2.28528 

22 

•  39  .    37687 

265.'542 

39694 

2.51921* 

41728 

2.39645 

43793 

2.28348 

21 

40  ;    37720 

2.  651  (19 

39727 

2.51715 

41763 

2.39449 

43828 

2.28167 

20 

41     .37754 

2.64875 

39761 

2.51502 

41797 

2.39253 

43862 

2.27987 

19 

42)    37787 

264642 

39795 

2.51289 

41831 

2.39058 

43897 

2.27806 

18 

43  i  .37820 

264410 

39829 

2.51076 

41865 

2.3S863 

43932 

2.27626 

17 

44      37S53 

264177 

39862 

2.50864 

.41899 

2.38668 

43966 

2.27447 

16 

45 

.37887 

2.63945 

39896 

2.50652 

.41933 

2.38473 

44001 

2.27267 

15 

16 

37920 

263714 

39930 

2.50440 

41963 

2.38279 

44036 

2.27088 

14 

47 

37953 

2  634*3 

39963 

2.502-29 

42002 

2.38084 

44071 

2.26909 

13 

48 

37986 

263252 

39997 

250018 

42036 

2.37891 

.44105 

2.26730 

12 

49 

.38020 

263021 

40031 

2.49807 

42070 

2.37697 

.44140 

2.26552 

11 

50 

.38053 

262791 

40065 

2.49597 

42105 

2.37504 

.44175 

2.26374 

10 

51 

.38086 

2  62561 

40098 

2.49386 

42139 

2.3731  1 

44210 

2.26196 

9 

52 

38120 

2  62£fi 

40132 

2.49177 

4-2173 

2.371  18 

44244 

2.26018 

8 

53 

38153 

262103 

40166 

24S967 

42207 

2.36925 

.44279 

2.25840 

7 

54 

.38186 

261874 

4021  K) 

2.48758 

4-2242 

2.36733 

.44314 

2.25663 

6 

55 

.3^220 

26IR46 

40-23-1 

2  48549 

42-276 

236541 

.44349 

225486 

6 

56 

.3-1253 

261418 

40267 

2.48340 

42310 

2.36349 

.44384 

225309 

4 

57 

38286 

261190 

41  1301 

248132 

42345 

2.36158 

44418 

2.25132 

3 

58 

.38320 

260963 

40335 

2  479'24 

.4-2379 

2.35967 

.44453 

2.24956 

2 

59 

.3^353 

260736 

.40369 

247716 

4-2413 

2.35776 

.44488 

2.24780 

1 

60 

.38386 

2.60509 

40403 

2.47509 

.42447 

2.35585 

.44523 

2.24604 

0 

IT 

Gotang. 

Tang. 

Cotang. 

Tang. 

Gotang. 

Tang. 

Cotang. 

Tang. 

M. 

|            69° 

68° 

67° 

66° 

2!)j!      TABLE     IV.        NATURAL     TANGEXTS     AND     COTAXGKX'I : 


i 

40 

1 

J5° 

a 

fto 

\ 

570 

1 

M. 

Tang. 

Cotang. 

Tang. 

Cotaug 

Tang. 

Cotang. 

Tung 

Cotaug. 

M. 

If 

744523 

2.24604 

.46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

60 

I 

.44558 

2.24428 

.46666 

2.14288 

.48809 

2.04879 

.50989 

1.961-20 

59 

2 

.44593 

2.24:452 

.467)12 

2.14125 

.48845 

2.04728 

.51026 

1.95979 

58 

3 

.44627 

2.24077 

.467!{7 

2.13963 

.48881 

2.04577 

.51063 

1.95838 

57 

4 

.44662 

2.23902 

46772 

2.13801 

.48917 

2.04426 

.51099 

1.95698 

56 

5 

.44697 

2.23727 

46808 

2.13639 

.48953 

2.04276 

.51136 

1.95557 

55 

6 

.44732 

2.23553 

.46>t43 

2.13477 

48989 

2.04125 

.51173 

1.95417 

54 

7 

.44767 

2.23378 

46879 

2.13316 

.49026 

2.03975 

51209 

1.95277 

53 

8 

.44802 

2.23204 

46914 

2.13154 

.49062 

2.03825 

.51246 

1.95137 

52 

9 

.44837 

2.23030 

46950 

2.12993 

.49098 

2.03675 

.51283 

1.94997 

51 

10 

.44872 

2.22857 

46985 

2.  12832 

.49134 

2.03526 

.61319 

1.94858 

50 

11 

.44907 

2.22683 

47021 

2.12671 

.49170 

203376 

51356 

1.94718 

49 

12 

.44942 

2.22510 

47056 

2.12511 

49206 

2.03227 

51393 

1.94579 

48 

13 

44977 

2.22337 

47092 

2.12350 

49242 

2.03078 

51430 

1.94440 

47 

14 

.45012 

2.22164 

47128 

2.12190 

.49273 

2.02929 

51467 

1.94301 

46 

15 

45047 

2.21992 

47163 

2.1203*1 

49315 

2.02780 

51503 

1.94162 

45 

16 

45082 

2.21  SI  9 

47199 

2.11871 

.49351 

2.02631 

51540 

1.94023 

44 

17 

45117 

2.21647 

47234 

2.11711 

.49387 

2.02483 

51577 

1.93866 

43 

18 

45152 

2.21475 

47270 

2.11552 

49423 

2.02:}:35 

51614 

1.93746 

42 

19 

45187 

2.21304 

47305 

2.11392 

.49459 

2.02187 

51651 

1.93608 

41 

20 

45222 

2.21  132 

47341 

2.11233 

.49495 

2.02039 

51688 

1.93470 

40 

21 

45257 

2.20961 

47377 

2.11075 

49532 

2.01891 

51724 

1.93332 

39 

22 

45292 

2.2079ft 

47412 

2.10916 

.49563 

2.01743 

51761 

1.93195 

38 

23 

45327 

2.20619 

.47443 

2.10758 

49604 

2.  01  596 

51793 

1.93007 

37 

24 

45362 

2.20449 

47483 

2.10600 

49640 

2.0144'J 

51835 

1.92020 

36 

25 

45397 

2.20278 

47519 

2.10442 

.49677 

2.01302 

51872 

1.92782 

35 

26 

45432 

2.20108 

47555 

2.10284 

49713 

2.01155 

51909 

1  92645 

34 

27 

45467 

2.19938 

47590 

2.10126 

49749 

2.01008 

51946 

i.  92508 

33 

28 

45502 

2.19769 

47626 

2.09969 

49786 

2.00862 

51983 

1.92371 

32 

29 

46538 

2.19599 

47662 

2.09311 

49822 

2.00715 

52020 

1.92235 

31 

30 

45673 

2.19430 

47698 

2.09654 

.49858 

2.00569 

52057 

1.92098 

30 

31 

45608 

2.19261 

47733 

2.09498 

49894 

2.00423 

52004 

1.91962 

29 

32 

45643 

2.1909-2 

47769 

2.09341 

49931 

2.00277 

52131 

1.91826 

28 

33 

45678 

2.18923 

47805 

2.09184 

49967 

2.00131 

52168 

1.91690 

27 

34 

45713 

2.18755 

47840 

2.09028 

50004 

1.999*6 

52205 

1.91554 

26 

35 

45748 

2.18587 

47876 

2.08872 

50(MO 

1.99H41 

52242 

1.91418 

25 

36 

45784 

2.18419 

47912 

2.08716 

50076 

1.99695 

62-279 

1.91282 

24 

37 

45819 

2.18251 

47948 

2.08560 

50113 

1.9955(1 

52316 

1.91147 

23 

38 

45854 

2.18084 

47984 

2.08405 

50149 

1.99406 

52353 

1.91012 

22 

39 

45889 

2.17916 

48019 

2.08250 

.50185 

1.99261 

52390 

1.90876 

21 

40 

45924 

2.17749 

48055 

2.0S094 

.50222 

1.99116 

52427 

1.90741 

20 

41 

45960 

2.17682 

48091 

2.07939 

.50258 

1.98972 

62464 

1.90607 

(9 

42 

45995 

2.17416 

48127 

2.07785 

50295 

.98828 

52501 

1.90472 

18 

43 

46030 

2.17249 

48163 

2.07630 

50331 

.98684 

52538 

1.90337 

17 

44 

46065 

2.17083 

48198 

2.07476 

60368 

.98540 

52575 

1.90203 

16 

45 

46101 

2.16917 

48234 

2.07321 

50404 

.98396 

52613 

1.90069 

15 

46 

46136 

2.16751 

48270 

2.07167 

50441 

.98253 

52650 

1.89935 

14 

47 

46171 

2.16585 

48306 

2.07014 

50477 

.98110 

52687 

1.89801 

13 

48 

46206 

2.16420 

48342 

2  06860 

50514 

.97966 

52724 

1.89667 

12 

49 

46242 

2  16255 

.48378 

2.06706 

.60550 

.97823 

52761 

1.89533 

11 

50 

46277 

2.16090 

.48414 

2.06553 

.50587 

.97681 

52798 

1.89400 

10 

51 

46312 

2.15925 

.48450 

2.06400 

50623 

.97538 

52836 

1.89266 

9 

52 

46348 

2.  15760 

.48486 

2.06247 

50660 

97395 

52873 

1.89133 

8 

53 

46383 

2.  15596 

48521 

2.06094 

50696 

.97253 

52910 

1.89000 

7 

54 

46418 

2.15432 

48557 

2.05942 

50733 

.97111 

.52947 

1.88867 

55 

46454 

2  15268 

48593 

2.05790 

.50769 

.96969 

529S5 

1.88734 

56 

46489 

215104 

48629 

2.05637 

50806 

.96.S27 

63022 

1.88602 

57 

46/525 

2.14940 

48665 

2.05485 

50843 

1.96685 

59159 

1.88469 

58 

46560 

2.14777 

48701 

2.05333 

50879 

.96544 

53096 

1.88337 

59 

46*95 

2.14614 

48737 

2.05182 

50916 

1.96402 

53134 

1.88205 

I 

60 

.46631 

2.14451 

48773 

2.05030 

50953 

.96'26I 

53171 

1.88073 

0 

M. 

Cutang. 

Tang. 

Cotang. 

Tang. 

Co  tang. 

Tang. 

Cotang. 

Tung. 

M. 

G 

[JO 

6 

1° 

6 

JO 

O! 

8° 

TARLI-;     IV.        NATURAL     TANGENTS     AND     COTANCKX 


380 

393 

300 

310 

M 

Tang 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang 

M. 

~o 

"63171 

1.88073 

55431 

1.80405 

.57735 

.73205 

60086 

1.66428 

6U 

63208 

1.87941 

55469 

1.802S1 

.57774 

.73089 

60126 

1.66318 

59 

63246 

1.87809 

55507 

1.8W58 

.57813 

.72973 

60165 

1.66209 

68 

53283 

.87677 

55545 

1.80034 

.57851- 

.72857 

60205 

1.66099 

67 

63320 

.87546 

55583 

1.79911 

.57890 

.72741 

60245 

1.65990 

56 

63358 

.87415 

65621 

1.79788 

.57929 

.726*5 

80384 

1.65881 

66 

63395 

.87283 

55659 

1.79665 

.57968 

.72509 

60324 

1.65772 

64 

63432 

.87152 

55697 

1.79542 

.58007 

.72393 

60364 

.65663 

53 

.63470 

.87021 

55736 

1.79419 

.58046 

.72278 

60403 

.65554 

62 

.63507 

.86891 

55774 

1.79296 

.58085 

.72163 

60443 

.65445 

51 

10 

63545 

.86760 

55812 

1.79174 

.58124 

.72047 

60483 

.65337 

60 

11 

53582 

.86630 

55850 

1.79051 

.58162 

.71932 

60522 

.65228 

49 

12 

53620 

.86499 

55888 

1.78929 

.58201 

.71817 

60562 

.65120 

48 

13 

63657 

.86369 

55926 

1.78807 

.58240 

.71702 

.60602 

.65011 

47 

14 

53694 

.86239 

55964 

1.78685 

.58279 

.71588 

.60642 

.64903 

46 

15 

.63732 

.86109 

56003 

1.78563 

.68318 

.71473 

.60681 

.64795 

45 

16 

63769 

.85979 

56041 

1.78441 

.58357 

.71358 

.60721 

.64687 

44 

17 

53807 

.85850 

56079 

1.78319 

.58396 

.71244 

60761 

.64579 

43 

18     53844 

.85720 

56117 

1.78198 

58435 

.71129 

60801 

.64471 

42 

19      53882 

.85591 

56156 

1.78077 

.58474 

.71015 

60841 

.64363 

41 

20     53920 

1.85462 

56194 

1.77955 

.58513 

.70901 

60881 

.64256 

40 

21      63957 

1.85333 

56232 

1.77834 

.58552 

.70787 

60921 

.64148 

39 

22      53995 

1.85204 

56270 

1.77713 

.68591 

.70673 

60960 

.64041 

38 

23      54032 

1.85075 

56309 

1.77592 

.58631 

.70560 

61000 

.63934 

37 

24      64070 

1.84946 

56347 

1.77471 

.58670 

.70446 

61040 

.63826 

36 

25      54107 

1.84818 

56385 

1.77351 

.58709 

.70332 

61080 

.63719 

36 

26     64145 

1.84689 

56424 

1.77230 

.58748 

.70219 

61120 

.63612 

34 

27     54183 

1.84561 

56462 

1.77110 

.58787 

.70106 

61160 

63505 

33 

28      54220 

1.84433 

56501 

1.76990 

.58826 

1.69992 

61200 

63398 

32 

29      54258 

1.84305 

56539 

1.76869 

.58865 

1.69879 

61240 

.63292 

31 

30 

64296 

1.84177 

.56577 

1.76749 

.58906 

1.69766 

61280 

1.63165 

30 

31 

64333 

1.84049 

56616 

1  76629 

.58944 

1.69653 

61320 

1.63079 

29 

32     64371 

.83922 

56654 

1.76510 

.58983 

1.69541 

.61360 

1.62972 

28 

33    .64409 

.83794 

56693 

1.76390 

.59022 

1.69428 

.61400 

1.62866 

27 

34     64446 

.83667 

56731 

1.76271 

.59061 

1.69316 

61440 

1.62760 

26 

35     64484 

.83540 

56769 

1.76151 

.59101 

1.69203 

61480 

1.62654 

06 

36     54522 

.83413 

56S08 

1.76032 

.69140 

1.69091 

61520 

1.62548 

24 

37     .64560 

.83286 

56S46 

1.75913 

.59179 

1.68979 

61561 

1.62442 

23 

38     54597 

.83159 

56885 

1.75794 

.59218 

.68866 

61601 

1.62336 

22 

39 

64635 

.83033 

56923 

1.75675 

.59258 

.68754 

61641 

1.62230 

21 

40 

64673 

.82906 

56962 

1.75556 

.59297 

.68643 

.61681 

1.62125 

20 

41      64711 

.82780 

57000 

1.75437 

.59336 

.68531 

.61721 

1.62019 

19 

42 

64748 

.82654 

57039 

1.75319 

.59376 

.68419 

.61761 

1.61914 

18 

43 

64786 

1.82528 

57078 

1.75200 

.59415 

.68308 

.61801 

1.61808 

17 

44 

64824 

1.82402 

57116 

1.75082 

.59454 

.68196 

.61842 

1.61703 

16 

45 

64862 

1.82276 

.57155 

1.74964 

.59494 

.68085 

.61882 

1.61598 

15 

M 

.64900 

1.82150 

.57193 

1.74846 

.59533 

.67974 

61922 

161493 

14 

47 

64933 

1.82025 

.57232 

1.74728 

.59573 

.67863 

61962 

1.61388 

13 

48 

.64975 

1.S1899 

.57271 

1.74610 

.59612 

.67752 

62003 

1.61283 

12 

it 

.65013 

1.81774 

.57309 

1.74492 

.59651 

.67641 

62043 

1.61179 

11 

5C 

.65051 

1.81649 

.57348 

1.74375 

.59691 

.67530 

62083 

1.61074 

10 

61 

55089 

1.81524 

.57386 

1.74257 

59730 

.67419 

62124 

1.60970 

9 

52 

.65127 

1.81399 

.57425 

1.74140 

59770 

.67309 

62164 

1.60865 

8 

53 

.55165 

1  81274 

57464 

1.74022 

.59809 

.67198 

62204 

1.60761 

7 

54 

55203 

1.81150 

57503 

1.73905 

.59849 

.67088 

.62245 

1.60657 

56 

55241 

1.81025 

.57541 

1.73788 

.59888 

.66978 

62285 

.60553 

66 

6527* 

1.80901 

.57580 

1.73671 

59928 

.66867 

.62325 

.60449 

57 

55317 

1.80777 

57619 

1.73555 

59967 

.66757 

62366 

.60345 

68 

55355 

1.80653 

57657 

173438 

60007 

.66647 

.62406 

.60241 

69 

55393 

1.80529 

57696 

1  73321 

.60046 

1.66538 

62446 

.60137 

60 

.55431 

1.80405 

.57735 

1.73205 

.60086 

.66428 

62487 

.60033 

M: 

Cotaug. 

Taug. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 

610 

600 

590 

680 

204 


TARLE     IV.        NATURAL     TANGENTS     AND     COTANGKXTS. 


i 

«o 

9 

13° 

3 

40 

1 

IC° 

M 

JUang. 

Cotaug. 

Taug. 

Cotaog. 

Tang. 

Cotaug. 

Taug. 

Cotang. 

M. 

~o 

.62487 

.60033 

.64941 

1.53986 

.67451 

1.43256 

.70021 

1.42816 

60 

I 

.62527 

.59930 

.64982 

1.53888 

.67493 

1.48163 

.70064 

.42726 

69 

2 

.62568 

.  59HJ&6 

.65024 

1.53791 

.67536 

1.48070 

.70107 

.42638 

63 

3 

.62608 

.59723 

.65065 

1.53693 

.67578 

1.47977 

.70151 

.42550 

67 

4 

.62649 

.59620 

.65106 

1.53595 

.67620 

1.47885 

.70194 

.42462 

66 

6 

.62689 

.69517 

.65148 

1.53497 

.67663 

1.47792 

.70238 

.42374 

66 

6 

.62730 

.69414 

.65189 

1.53400 

.67705 

1.47699 

.70281 

.42286 

64 

7 

.62770 

1.59311 

.65231 

1.633(12 

.67748 

1.47607 

70325 

1.42198 

63 

8 

.62811 

1.59208 

65272 

1.63205 

.67790 

1.47514 

.70368 

1.42110 

62 

9 

.62852 

1.69105 

65314 

1.63107 

.67832 

1.47422 

.70412 

142022 

61 

10 

.62892 

1.59002 

.65355 

1.53010 

.67875 

1.47330 

70455 

1.41934 

60 

11 

.62933 

1.68900 

65397 

1.62913 

.67917 

1.47238 

70499 

1.41847 

49 

12 

.62973 

.58797 

65438 

1.52816 

.67960 

1.47146 

.70542 

1.41759 

48 

13 

.63014 

1.58695 

65480 

1.62719 

.68002 

1.47053 

.70586 

1.41672 

47 

14 

.63055 

1.58593 

65521 

1.52622 

.68045 

1.46962 

.70629 

1.41684 

46 

16 

.63095 

1.68490 

65563 

1.62525 

.68088 

1.46870 

.70673 

1.41497 

46 

16 

.63136 

1.68388 

65604 

1.52429 

.68130 

1.46778 

.70717 

1.41409 

44 

17 

.63177 

1.58286 

65646 

1.52332 

.68173 

1.46686 

.70760 

1.41322 

43 

18 

.63217 

1.58184 

65688 

1.62235 

.68215 

1.46595 

.70804 

1.41235 

42 

19 

.63258 

.58083 

65729 

1.52139 

.68258 

1.46503 

.70848 

1.41148 

41 

20 

.63299 

1.67981 

65771 

1.52043 

68301 

1.46411 

.70891 

1.41061 

40 

21 

.63340 

1.57879 

.65813 

1.61946 

68343 

1.46320 

.70935 

1.40974 

39 

22 

.63330 

1.67778 

.65854 

1.61850 

.68386 

1.46229 

70979 

1.40887 

3S 

23 

.63421 

.67676 

65896 

1.61764 

.68429 

1.46137 

71023 

1.40800 

37 

24 

.63462 

.67575 

65938 

1.61658 

.68471 

1.46046 

.71066 

.40714 

36 

26 

.63503 

.67474 

65980 

1.61662 

.68514 

1.45955 

.71110 

1.40627 

35 

26 

63544 

.67372 

66021 

1.61466 

.68567 

1.45864 

71164 

1.40540 

34 

27 

.63584 

.67271 

66063 

1.61370 

.68600 

1.45773 

.71198 

1.40454 

33 

28 

.63626 

.67170 

66106 

1.61276 

.68642 

1.45682 

.71242 

1.40367 

32 

29 

.63666 

.67069 

.66147 

1.61179 

.68685 

1.45592 

.71286 

1.40281 

31 

30 

.63707 

.66969 

66189 

1.61084 

.68723 

1.45501 

.71329 

1.40196 

30 

31 

.63748 

.66868 

66230 

1.60988 

.68771 

1.45410 

.71373 

1.40109 

29 

32 

.63789 

.66767 

66272 

1.60893 

.68814 

1.45320 

.71417 

1.40022 

28 

33 

.63830 

.56667 

66314 

1.60797 

.68857 

1.45229 

.71461 

1.39936 

27 

24 

.63871 

.66566 

66356 

1.60702 

.68900 

1.45139 

.71606 

1.39850 

26 

35 

.63912 

.56466 

66398 

1.50607 

.68942 

1.45049 

.71649 

1.39764 

26 

36 

.63953 

.56366 

66440 

1.60512 

.68985 

1.44958 

.71593 

1.39679 

24 

37 

.63994 

.56265 

664  S2 

1.50417 

.69028 

1.44868 

.71637 

1.39593 

23 

38 

.64035 

.66165 

66524 

1.50322 

.69071 

1.44778 

.71681 

1.39507 

23 

39 

.64076 

.5606o 

66566 

1.50228 

.69114 

1.44688 

71725 

1.39421 

21 

40 

.64117 

.55966 

66608 

150133 

.69157 

1.44598 

71769 

1.39336 

20 

41 

.64158 

.55866 

66650 

1.50038 

.69200 

1.44508 

71813 

1.39250 

19 

42 

.64199 

.65766 

666D2 

1.49944 

.69243 

1.44418 

.71867 

1.39165 

18 

43 

.64240 

.65666 

.66734 

1  49849 

.692S6 

1.44329 

.71901 

1.39079 

17 

44 

.64281 

.65567 

66776 

1  49755 

.69329 

1.44239 

.71946 

1.38994 

16 

45 

.64322 

.66467 

.66818 

1.49661 

.69372 

1.44149 

.71990 

1.38909 

16 

46 

.64363 

.65368 

.66860 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14 

«r 

.64404 

.55269 

.66902 

1.49472 

.69459 

1.43970 

.72078 

1.38738 

13 

48 

.64446 

.65170 

.66944 

1.49378 

.69502 

1.43881 

.72122 

1.38653 

11 

49 

.64487 

.65071 

.66986 

1.49284 

.69545 

1.43792 

.72167 

1.38568 

11 

60 

.64528 

.54972 

.67028 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

10 

61 

.64569 

.54873 

.67071 

1.49097 

.69631 

1.43614 

72255 

1.38399 

9 

62 

.64610 

.54774 

.67113 

1.49003 

.69675 

1.43525 

72299 

1.38314 

8 

63 

.64652 

.54675 

.67155 

1.48909 

.69718 

1.43436 

.72344 

1.38229 

7 

54 

.64693 

.64576 

.67197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

6 

66 

.64734 

.64478 

.67239 

1.48722 

.69804 

1.43258 

.72432 

1.38060 

6 

66 

.64775 

.54379 

.67282 

1.48629 

.69847 

1.43169 

.72477 

1.37976 

4 

67 

.64817 

.64281 

.67324 

1.48536 

.69^91 

1.43080 

.72521 

1.37891 

3 

68 

.64868 

1.64183 

.67366 

1.48442 

.69934 

1.42992 

.72565 

1.37807 

a 

69 

.64899 

1.64085 

.67409 

1.48349 

.69977 

1.42903 

.72610 

1.37722 

•  i 

60 

.64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

.72654 

1.37638 

0 

IT 

Ootaiig. 

Tang. 

Cotang. 

-TS5T 

Cotaug. 

Tang. 

Cotaug. 

?*"«• 

s: 

6 

70 

54 

JO 

5 

BO 

5' 

10 

TABLE     IV.        NATURAL     TANGENTS     AND     COTANC.EN' 


3 

GO 

3 

T° 

3 

8° 

31 

DO 

M. 

Tmng. 

Cotang. 

Tang. 

Cotang. 

Tung. 

Cotang. 

Tang. 

Cotang. 

M. 

~0 

.72654 

.37638 

.75355 

.32704 

.78129 

1.27994 

.80978 

1.23490 

60 

.72699 

.37554 

.75401 

.32624 

.78175 

1.27917 

.81027 

1.23416 

69 

.72743 

.37470 

.75447 

.32544 

.78222 

1.27841 

.81075 

1.23343 

68 

.72788 

.37386 

.75492 

.32464 

.78269 

1.27764 

.81123 

1.23270 

67 

.72832 

.37302 

.75538 

.32384 

.78316 

1.27683 

.81171 

1.23196 

56 

.72877 

.37218 

.75584 

.32304 

.78363 

1.27611 

.81220 

1.23123 

55 

.72921 

.37134 

.75629 

.32224 

.78410 

1.27535 

.81268 

1.23050 

64 

.72966 

.37050 

.75675 

.32144 

.78457 

1.27458 

.81316 

1.22977 

63 

.73010 

.36967 

.75721 

.32064 

.78504 

1.27382 

.81364 

1.22904 

62 

.73055 

.36883 

.75767 

31984 

.78551 

1.27306 

.814(3 

1.22831 

51 

10 

.73100 

.36800 

.75812 

.31904 

.78598 

1.27230 

.81461 

1.22758 

60 

11 

.73144 

.36716 

.75858 

.31825 

.78645 

1.27i53 

.81510 

1.22685 

49 

12 

.73189 

.36633 

.75904 

.31745 

.78692 

1.27077 

.81558 

1.22612 

48 

13 

.73234 

.36549 

.75950 

.31666 

.78739 

1.27001 

.81606 

1.22539 

47 

14 

.73278 

.36466 

.75996 

.31586 

.78786 

1.26925 

.81655 

1.22467 

46 

15 

.73323 

.36383 

.76042 

.31507 

.78834 

1.26849 

.81703 

1.22394 

45 

16 

.73368 

.36300 

.76088 

.31427 

.78881 

1.26774 

.81752 

1.22321 

44 

17 

.73413 

.36217 

.76134 

.31348 

.78928 

1.26698 

.81800 

1.22249 

43 

18 

.73457 

.36134 

.76180 

.31269 

.78975 

1.26622 

.81849 

1.22176 

42 

19 

.73502 

.36051 

.76226 

.31190 

.79022 

1.26546 

.81898 

1.22104 

41 

20 

.73547 

.35968 

.76272 

.31110 

.79070 

1.26471 

.81946 

1.22031 

40 

21 

.73592 

.35885 

.76318 

.31031 

.79117 

1.26395 

.81995 

1.21959 

39 

22 

.73637 

.35802 

.76364 

.30952 

.79164 

1.26319 

.82044 

1.21886 

38 

23 

.73681 

.35719 

.76410 

.30873 

.79212 

1.26244 

.82092 

1.21814 

37 

24 

.73726 

.35637 

.76456 

.30795 

.79259 

1.26169 

.82141 

1.21742 

36 

25 

.73771 

.35554 

.76502 

.30716 

79306 

1.26093 

.82190 

1.21670 

36 

26 

.73816 

.35472 

.76548 

.30637 

.79354 

1.26018 

82238 

1.21598 

34 

27 

.73861 

.35389 

.76594 

.30558 

.79401 

1.25943 

.82287 

1.21526 

33 

28 

.73906 

.35307 

.76640 

.30480 

.79449 

1.25867 

.82336 

1.21454 

32 

29 

.73951 

.35224 

.76686 

.30401 

.79496 

1.25792 

.82385 

1.21382 

31 

30 

.73U96 

.35142 

.76733 

.30323 

.79544 

1.25717 

.82434 

1.21310 

30 

31 

74041 

1.35060 

.76779 

.30244 

.79591 

1.25642 

.82483 

1.21238 

29 

32 

.74086 

1.34978 

.76825 

.30166 

.79639 

1.25567 

82531 

1.21166 

28 

33 

74131 

1.34896 

.76871 

.30087 

.79686 

1.25492 

.82580 

1.21094 

27 

34 

74170 

1.34814 

.76918 

.30009 

.79734 

1.25417 

82629 

1.21023 

26 

35 

.74221 

1.34732 

.76964 

.29931 

.79781 

1.25343 

.82678 

1.20951 

26 

36 

.74267 

1.34650 

.77010 

1.29853 

.79829 

1.25268 

.82727 

1.20879 

24 

37 

74312 

1.34568 

.77057 

1.29775 

.79877 

1.25193 

.82776 

1.20808 

23 

38 

74357 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

.82825 

1.20736 

22 

39 

.74402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

1.20665 

21 

40 

.74447 

1.34323 

.77196 

1.29541 

.80020 

1.24969 

82923 

1.20593 

20 

41 

.74492 

1.34242 

.77242 

1.29463 

.80067 

1.24895 

.82972 

1.20522 

19 

42 

.74538 

1.34160 

.77289 

1.29385 

.80116 

1.24820 

.83022 

1.20451 

18 

43 

.74583 

1.34079 

.77335 

1.29307 

,80163 

1.24746 

.83071 

1.20379 

17 

44 

.74628 

1.33998 

.77382 

1.29229 

.80211 

1.24672 

.83120 

1.20308 

16 

45 

.74674 

1.33916 

.77428 

1.29152 

,80258 

1.24597 

.83169 

1.20237 

16 

46 

74719 

1.33835 

.77476 

1.29074 

.80306 

1.24523 

.83218 

1.20166 

14 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

1.28919 

.80402 

1.24375 

.83317 

1.20024 

12 

49 

.74855 

1.33592 

.77615 

1.28842 

.80450 

1.24301 

.83366 

1.19953 

11 

60 

.74900 

1.33511 

.77661 

.28764 

.80498 

1.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

.28687 

.80546 

1.24153 

.83465 

1.19811 

9 

62 

.74991 

1.33349 

.77754 

.28610 

.80594 

1.24079 

.83514 

1.19740 

8 

63 

.75037 

1.33268 

.77801 

.28533 

.80642 

1.24005 

.83564 

1.19669 

7 

64 

.76082 

1.33187 

.77848 

.28456 

.80690 

1.23931 

.83613 

1.19599 

6 

65 

.75128 

1.33107 

.77895 

.28379 

.80738 

1.23858 

.83662 

1.19528 

6 

66 

.75173 

1.33026 

.77941 

.28302 

.80786 

1.23784 

.83712 

1.19457 

4 

57 

.75219 

1.32946 

.77988 

1.28225 

.80834 

1.23710 

.83761 

1.19387 

3 

68 

.76264 

1.32865 

.78035 

1.28148 

.80882 

1.23637 

.83811 

1.19316 

2 

69 

.75310 

1.32785 

.78082 

1.28071 

.80930 

1.23563 

.83860 

1.19246 

1 

60 

.75355 

1.32704 

.78129 

.27994 

.80978 

1.23490 

.83910 

1.19175 

0 

M. 

Ootang. 

Tang. 

Cotaiig. 

Tang. 

Cotaug. 

Tang. 

Cotang. 

Tang. 

M. 

e 

30 

0 

58° 

5 

1° 

5 

QO 

I'lKI      TAKLE     IV.        NATURAL     TANGENTS     AND     COTANGENTS. 


4O° 

410 

4»o 

430 

M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

~0 

.83910 

1.19176 

.86929 

.15037 

.90040 

"  .11061 

.93259 

1.07237 

60 

j 

.83960 

.19105 

.86980 

.14969 

.90093 

.10996 

.93306 

.07174 

59 

.84009 

.19035 

.87031 

.14902 

.90146 

.10931 

.93360 

.07112 

58 

.84069 

.18964 

.87082 

.14834 

.90199 

.10867 

93415 

.07049 

57 

.84108 

.18894 

.87133 

.14767 

.90251 

.10602 

93469 

.06987 

56 

.84158 

.18824 

.87184 

.14699 

.903(4 

.10737 

93524 

.06925 

55 

.84208 

.18754 

.87236 

.14632 

.90357 

.10672 

93578 

.06862 

54 

.84258 

.18684 

.87287 

.14565 

.90410 

.10607 

93633 

06800 

53 

.84307 

.18614 

.87338 

.14498 

.90463 

.10543 

93688 

.06738 

52 

.84357 

.18544 

.87389 

.14430 

.90516 

.10478 

93742 

.06676 

51 

10 

.84407 

.18474 

.87441 

14363 

.90569 

.10414 

93797 

.06613 

50 

11 

.84457 

.18404 

.87492 

.14296 

.90621 

.10349 

93852 

.06551 

49 

12 

.84507 

.18334 

.87543 

.14229 

.90674 

.10285 

.93906 

.06489 

48 

13 

.84556 

.18264 

.87595 

.14162 

.90727 

.10220 

.93961 

.06427 

47 

14 

.84606 

.18194 

.87646 

.14095 

.90781 

.10156 

.94016 

.06365 

46 

15 

.84656 

.18125 

.87693 

.14028 

.90834 

.10091 

.94071 

.06303 

45 

16 

.84706 

.18055 

.87749 

.13961 

.90887 

.10027 

.94125 

.06241 

44 

17 

84756 

.17986 

.87801 

.13894 

.90940 

.09963 

.94180 

.06179 

43 

18 

.84806 

.17916 

87852 

.13828 

.90993 

1.09899 

94235 

.06117 

42 

19 

.84856 

.17846 

87904 

.13761 

.91046 

1.09834 

94290 

.06056 

41 

20 

84906 

.17777 

87955 

.13694 

.91099 

1.09770 

.94345 

.05994 

40 

31 

.84956 

.17708 

.88007 

.13627 

.91153 

1.09706 

94400 

.05932 

39 

22 

.85006 

.17638 

.88059 

.13561 

.91206 

1.09642 

94455 

.05870 

38 

23 

.85057 

.17569 

.88110 

.13494 

.91259 

1.09578 

.94510 

.05809 

37 

24 

.85107 

.17500 

.88162 

.13428 

.91313 

1.09514 

.94565 

.05747 

36 

25 

.85157 

.17430 

.88214 

.13361 

.91366 

1.09450 

94620 

.05685 

36 

26 

85207 

.17361 

.88265 

.13295 

.91419 

1.09386 

.94676 

.05624 

34 

27 

.85257 

.17292 

.88317 

.13228 

.91473 

1.09322 

94731 

.05562 

33 

28 

.85308 

.17223 

.883^9 

.13162 

.91526 

1.09258 

.94786 

.05601 

32 

29 

85358 

.17154 

.88421 

.13096 

.91580 

1.09195 

.94341 

.05439 

31 

30 

.85408 

.17085 

.88473 

.13029 

.91633 

1.09131 

.94896 

1.05378 

30 

31 

.86458 

.17016 

.88524 

.12963 

.91687 

1.09067 

94952 

.05317 

29 

32 

.85509 

.16947 

.88576 

.12897 

.91740 

1.09003 

95007 

.05255 

28 

33 

.85559 

.16878 

.88628 

.12831 

.91794 

1.08940 

95062 

.05194 

27 

34 

.85609 

.16809 

.88680 

.12765 

.91847 

1.08876 

.95118 

.05133 

26 

35 

.85660 

.16741 

.88732 

.12699 

.91901 

1.08813 

.95173 

.05072 

25 

36 

.85710 

.16672 

.88784 

.12633 

.91955 

1.08749 

95229 

.05010 

24 

37 

.85761 

.16603 

.88836 

.12567 

.92008 

1.08686 

.95234 

.04949 

23 

38 

.85811 

.16535 

.88888 

.12501 

.92062 

1.08622 

.95340 

1.04888 

22 

39 

.85862 

.16466 

.88940 

.12435 

.92116 

1.08559 

95395 

1.04827 

21 

40 

85912 

.16398 

.88992 

.12369 

.92170 

1.08496 

95451 

1.04766 

20 

41 

.85963 

.16329 

.89045 

.12303 

.92224 

1.08432 

.95506 

1.04705 

19 

42 

.86014 

.16261 

.89097 

1.12238 

.92277 

1.08369 

95562 

1.04644 

18 

43 

.86064 

.16192 

.89149 

.12172 

.92331 

.08306 

.95618 

1.04583 

17 

44 

.86115 

.16124 

.89201 

.12106 

.92385 

1.08243 

.95673 

1.04522 

16 

45 

.86166 

.16056 

.89253 

.12041 

.92439 

1.08179 

.95729 

1.04461 

15 

46 

.86216 

.15987 

.89306 

.11975 

.92493 

1.08116 

95785 

1.04401 

14 

47 

.86267 

.15919 

89358 

.11909 

.92547 

1.08053 

.95841 

1.04340 

13 

48 

.86318 

.15851 

.89410 

.11844 

.92601 

1.07990 

95897 

1.04279 

12 

49 

.86368 

.15783 

.89463 

.11778 

.92655 

1.07927 

95952 

1.04218 

11 

50 

.86419 

.15715 

.89515 

.11713 

.92709 

1.07864 

.96008 

1.04158 

10 

51 

.86470 

.15647 

.89567 

.11648 

.92763 

1.07801 

.96064 

1.04097 

52 

.86521 

.15579 

.89620 

.11582 

.92817 

1.07738 

96120 

1.04036 

53 

.86572 

.15511 

.89672 

.11517 

.92872 

1.07676 

.96176 

1.03976 

54 

.86623 

.15443 

.89725 

.11452 

.92926 

1.07613 

.96232 

1.03915 

55 

.86674 

.15375 

.89777 

.11387 

.92980 

1.07550 

.96288 

1.03855 

56 

.86725 

.15308 

.89830 

.11321 

.93034 

1.07487 

.96344 

1.03794 

57 

.86776 

.15240 

.89883 

.11256 

.93088 

1.07425 

.96400 

1.03734 

58 

.86827 

.15172 

.89935 

.11191 

.93143 

1.07362 

.96457 

1.03674 

59 

.86878 

.15104 

.89988 

.11126 

.93197 

.07299 

.96513 

1.03613 

60 

.86929 

1.15037 

.90040 

.11061 

.93252 

1.07237 

.96569 

1.03553 

IT 

Gotaug. 

Tang. 

Cotaug. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M~ 

490 

48° 

470 

46° 

TABLE     IV.        NATURAL     TANGENTS     AND     COTANGENTS. 


4 

4° 

1             4' 

1° 

4^ 

1° 

M. 

Tans. 

Cotang. 

M. 

M. 

Tang. 

Cotang. 

M. 

M. 

Tang. 

Cotang. 

M. 

0 

.96569 

1.03553 

HO 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

1 

.9(5625 

1.03493 

59 

21 

.97756 

1.02295 

39 

41 

.98901 

1.01112 

19 

2 

.96681 

1.03433 

58 

22 

.97813 

1.02236 

38 

42 

.98958 

1.01053 

18 

3 

.96738 

1.03372 

57 

23 

.97870 

1.02176 

37 

43 

.99016 

1.00994 

17 

4 

.96794 

1.03312 

56 

24 

.97927 

1.02117 

36 

44 

.99073 

1.00935 

16 

5 

.96850 

1.03252 

55 

25 

.97984 

1.02057 

85 

45 

.99131 

1.00876 

15 

6 

.96907 

1.03192 

54 

213 

.98041 

1.01998 

34 

46 

.99189 

1.00818 

14 

.96963 

1.03132 

53 

27 

.98098 

1.01939 

33 

47 

.99247 

1.00759 

13 

8 

.97020 

1.03072 

52 

28 

.98155 

1.01879 

32 

48 

.99304 

1.00701 

12 

9 

.97076 

1.03012 

51 

29 

.98213 

1.01820 

31 

49 

.99362 

1.00642 

11 

10 

.97133 

1.02952 

50 

30 

.98270 

.01761 

30 

50 

.99420 

1.00583 

10 

11 

.97189 

1.02892 

49 

31 

.98327 

.01702 

M 

51 

.99478 

.00525 

9 

12 

.97246 

1.02832 

48 

32 

.98384 

.01642 

28 

52 

.99536 

.00467 

8 

13 

.97302 

1.02772 

47 

33 

.98441 

.01583 

27 

53 

.99594 

.00408 

7 

14 

.97359 

1.02713 

46 

34 

.98499 

.01524 

26 

54 

.99652 

.00350 

6 

15 

.97416 

1.02653 

45 

35 

.98556 

.01465 

25 

55 

.99710 

.00291 

5 

16 

.97472 

1.02593 

44 

3fi 

.98613 

1.01406 

?4 

56 

.99768 

.00233 

4 

17 

.97529 

1.02533 

« 

37 

.98671 

1.01347 

23 

57 

.99826 

.00175 

3 

18 

.97586 

1.02474 

42 

38 

.98728 

1.01288 

23 

58 

.99884 

.00116 

2 

19 

.97643 

1.02414 

41 

39 

.98786 

1.01229 

21 

59 

.99942 

.00058 

1 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

60 

1.00000 

1.00000 

0 

M. 

Cotang. 

Tang. 

31. 

M, 

Cotang. 

Tang. 

M. 

M. 

Cotang. 

Tang. 

M. 

4 

5° 

4. 

>° 

41 

i° 

TABLE  V. 

CUBIC  YARDS  PER  100  FEET.    SLOPES  %:!;%:!; 
1  :  1;  iVu  :  1;  2  :  1;  3  :  1. 


299 


loo 


TABLE  V. — CUBIC   YARDS    PER   100  FEET.      SLOPES    */£    :  1. 


Depth 

Base 
12 

Base 
H 

Base 
16 

Base 
18 

Base 
22 

Base 
24 

Base 
26 

Baso 
20 

1 

45 

53 

60 

68 

82 

90 

97 

105 

2 

93 

107 

122 

137 

167 

181 

196 

211 

3 

142 

163 

186 

208 

253 

275 

297 

319 

4 

193 

222 

252 

281 

341 

370 

400 

430 

5 

245 

282 

319 

356 

431 

468 

505 

542 

6 

300 

344 

389 

433 

522 

567 

611 

656 

7 

356 

408 

460 

512 

616 

668 

719 

771 

8 

415 

474 

533 

593 

711 

770 

830 

889 

9 

475 

542 

608 

675 

808 

875 

942 

1008 

10 

537 

611 

685 

759 

907 

981 

1056 

1130 

11 

601 

682 

764 

845 

1008 

1090 

1171 

1253 

12 

667 

756 

844 

933 

1111 

1200 

1289 

1378 

13 

734 

831 

926 

1023 

1216 

1312 

1408 

1505 

14 

804 

907 

1010 

1115 

1322 

1426 

1530 

1633 

15 

875 

986 

1096 

1208 

1431 

1542 

1653 

1764 

16 

948 

1067 

1184 

1304 

1541 

1659 

1778 

1896 

17 

1023 

1149 

1274 

1401 

1653 

1779 

1905 

2031 

18 

1100 

1233 

1366 

1500 

1767 

1900 

2033 

2167 

19 

1179 

1319 

1460 

1601 

1882 

2023 

2164 

2305 

^0 

1259 

1407 

1555 

1704 

2000 

2148 

2296 

2444 

21 

1342 

1497 

1653 

1808 

2119 

2275 

2431 

2586 

22 

1426 

1589 

1752 

1915 

2241 

2404 

2567 

S730 

23 

1512 

1682 

1&53 

2023 

2364 

2534 

2705 

2875 

24 

1600 

1778 

1955 

2133 

2489 

2667 

2844 

3022 

25 

1690 

1875 

2060 

2245 

2616 

2801 

2986 

3171 

26 

1781 

1974 

2166 

2359 

2744 

2937 

3130 

3322 

27 

1875 

2075 

2274 

2475 

2875 

3075 

3275 

3475 

28 

1970 

2178 

2384 

2593 

3007 

3215 

3422 

3630 

29 

2068 

2282 

2496 

2712 

3142 

asse 

3571 

3786 

30 

2167 

2389 

2610 

2&S3 

3278 

3500 

3722 

3944 

31 

2268 

2497 

2726 

2956 

3416 

3645 

3875 

4105 

32 

2370 

2607 

2844 

3081 

3556 

3793 

4030 

4267 

33 

2475 

2719 

2964 

3208 

3697 

3942 

4186 

4431 

34 

2581 

2833 

3085 

3337 

3841 

4093 

4344 

4596 

35 

2690 

2949 

3208 

3468 

3986 

4245 

4505 

4764 

36 

2800 

3067 

3333 

3600 

4133 

4400 

4667 

4933 

37 

2912 

3186 

3460 

3734 

4282 

4556 

4831 

5105 

88 

3026 

3307 

3589 

3870 

4433 

4715 

4996 

5278 

39 

3142 

3431 

3719 

4008 

4586 

4875 

5164 

5453 

40 

3259 

3556 

3852 

4148 

4741 

5037 

5333 

5630 

41 

3379 

3682 

3986 

4290 

4897 

5201 

5505 

5808 

42 

3500 

3811 

4122 

4433 

5056 

5367 

5678 

5989 

43 

3623 

3942 

4260 

4579 

5216 

5534 

5853 

6171 

44 

3748 

4074 

4400 

4726 

5378 

5704 

6030 

6356 

45 

3875 

4208 

4541 

4875 

5542 

5875 

6208 

6542 

46 

4004 

4344 

4684 

5026 

5707 

6048 

6389 

6730 

47 

4134 

4482 

4830 

5179 

5875 

6223 

6571 

6919 

48 

4267 

4622 

4978 

5333 

6044 

6400 

6756 

7111 

49 

4401 

4764 

5127 

5490 

6216 

6579 

6942 

7305 

50 

4537 

4907 

5278 

5648 

6389  , 

6759 

7130 

7500 

51 

4675 

5053 

5430 

5808 

6564 

6942 

7319 

7697 

52 

4815 

5200 

5584 

5970 

6741 

7126 

7511 

7896 

53 

4956 

5349 

5741 

6134 

6919 

7312 

7705 

8097 

54 

5100 

5500 

5900 

6300 

7100 

7500 

7900 

&300 

55 

5245 

5653 

6060 

6468 

7282 

7690 

8097 

8505 

56 

5393 

5807 

6222 

6637 

7467 

7881 

8296 

8711 

57 

5542 

5964 

6386 

6808 

7653 

8075 

8497 

8919 

58 

5693 

6122 

6552 

6981 

7841  : 

8270 

8700 

9130 

59 

5845 

6282 

6719 

7156   : 

8031 

8468 

8905 

9342 

60 

6000 

6444 

6889 

7333 

8222 

8667 

9111 

9556 

TABLE  V. — CUBIC   YARDS    PER    100   FEET.      SLOPES 


301 


Depth 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

12 

14    16 

18 

22 

24    26 

23 

j 

46 

54 

61 

09 

83 

91 

98 

106 

g 

96 

111 

126 

141 

170 

185 

200 

215 

0 

150 

17'2 

194 

217 

261 

283 

306 

328 

4 

207 

237 

267 

296 

356 

385 

415 

444 

5 

269 

306 

343 

380 

454 

491 

528 

565 

6 

333 

378 

422 

467 

556 

600 

644 

689 

7 

402 

454 

506 

557 

661 

713 

765 

317 

8 

474 

533 

593 

652 

770 

830 

889 

948 

9 

550 

617 

683 

750 

883 

950 

1017 

1083 

10 

630 

704 

778 

852 

1000 

1074 

1148 

1222 

11 

713 

794 

876 

957 

1120 

1202 

1283 

1365 

12 

800 

889 

978 

1067 

1244 

1333 

1422 

1511 

13 

891 

987 

1083 

1180 

1372 

1469 

1565 

1661 

14 

985 

1089 

1193 

1296 

1504 

1607 

1711 

1815 

15 

1083 

1194 

1306 

1417 

1639 

1750 

1861 

1972  . 

16 

1185 

1304 

1422 

1541 

1779 

1896 

2015 

2133 

17 

1291 

1417 

1543 

1669 

1920 

2046 

2172  • 

2298 

18 

1400 

1533 

1667 

1800 

2067 

2200 

2333 

2467 

19 

1513 

1654 

1794 

1935 

2217 

2357 

2498 

2639 

20 

1630 

1778 

1926 

2074 

2370 

2519 

2667 

2815 

21 

1750 

1906 

2061 

2217 

2528 

2683 

2839 

2994 

22 

1874 

2037 

2200 

2363 

2689 

2852 

3015 

3178 

23 

2002 

2172 

2343 

2513 

2854 

3024 

3194 

3365 

24 

2133 

2311 

2489 

2667 

3022 

3200 

3378 

3556 

25 

2269 

2454 

2639 

2824 

3194 

3380 

3565 

3750 

26 

2407 

2600 

2793 

2985 

3370 

&5C3 

3756 

3948 

27 

2550 

2750 

2950 

3150 

&550 

3750 

3950 

4151 

28 

2696 

2904 

3111 

3319 

3733 

3941 

4148 

4356 

29 

2846 

3061 

3276 

3491 

3920 

4135 

4350 

4565 

30 

3000 

3222 

3444 

3667 

4111 

4333 

4556 

4778 

31 

3157 

3387 

3617 

3846 

4306 

4535 

4765 

4994 

32 

3319 

3556 

3793 

4030 

4504 

4741 

4978 

5215 

33 

3483 

3728 

3972 

4217 

4706 

4950 

5194 

5439 

34 

3652 

3904 

4156 

4407 

4911 

5163 

5415 

5667 

35 

3824 

4083 

4343 

4602 

5120 

5380 

5639 

5898 

36 

4000 

4267 

4533 

4800 

5333 

5600 

5867 

6133 

37 

4180 

4454 

4728 

5002 

5550 

5824 

6098 

6372 

38 

4363 

4644 

4926 

5207 

5770 

6052 

6333 

6615 

39 

4550 

4839 

5128 

5417 

5994 

6283 

6572 

6861 

40 

4741 

5037 

5333 

5630 

6222 

6519- 

6815 

7111 

'  41 

4935 

5239 

5543 

5846 

6454 

6757 

7061 

7365 

42 

5133 

5444 

5756 

6067 

6689 

7000 

7311 

7622 

43 

5335 

5654 

5972 

6291 

6928 

7246 

7565 

7883 

44 

5541 

5867 

6193 

6519 

7170 

7496 

7823 

8148 

45 

5750 

6083 

6417 

6750 

7417 

7750 

8083 

8417 

46 

5963 

6304 

6644 

6985 

7667 

8007 

8348 

8689 

47 

6180 

6528 

6876 

7224 

7920 

8269 

8617 

8965 

48 
49 

6400 
6624 

6756 

6987 

7111 
7350 

7467 
7713 

ffi 

m 

50 

6852 

7222 

7593 

7963 

87G4 

9074 

9444 

9815 

51 

7083 

7461 

7839 

8217 

8972 

9350 

9728 

10106 

52 

7319 

7704 

8089 

8474 

9244 

9630 

10015 

10400 

53 

7557 

7950  I  8343 

8735    9520 

9913 

10306 

10698 

54 

7800 

8200 

8600 

9000 

9800 

10200 

10600 

11000 

55 

8046 

8454 

8861 

9269 

10083 

10491 

10898 

11306 

56 

8296 

8711 

9126 

9541 

10370 

10785 

11200 

11615 

57 

8550 

8972 

9394 

9817 

10661 

11083 

11506 

11928 

58 

8807 

9237 

9667 

10096 

10956 

11385  !  11815 

12244 

59 

9069 

9506 

9943 

10380 

11254 

11691  |  12128 

12565 

60 

9333 

9778 

10222 

10667 

11556 

12000  |  12444   12889 

302 


TABLE   V. CUBIC    YARDS    PER    100    FEET.       SLOPES    1    :  1. 


Depth 

Base 
12 

Base  j  Base 
14    16 

Base   Base 
18    20 

Base   Base 
28    30 

Base 
32 

1 

48 

56 

63  !   70     78 

107     115 

122 

2 

104 

119 

133     148  j   163 

222  1   237 

252 

3 

167 

189 

211     233 

256- 

344    367 

889 

4 

237 

267 

296    326 

356 

474 

504 

533 

5 

315 

352 

389 

426 

463 

611 

648 

685 

6 

loo 

444 

489 

533 

578 

756 

800 

844 

7 

493 

544 

596 

648 

700 

907 

959 

1011 

6 

593 

652 

711 

770 

830 

1067 

1126 

1185 

9 

700 

767 

833 

900 

967 

1233 

1300 

1367 

10 

815 

889 

963 

1037 

1111 

1407 

1481 

1556 

11 

937 

1019 

1100 

1181 

1263 

1589 

1670 

1752 

12 

1067 

1156 

1244 

1333 

1422 

1778 

1867 

1956 

13 

1204 

1300 

1396 

1493 

1589 

1974 

2070 

2167 

14 

1348 

1452 

1556 

1659 

1763 

2178 

2281 

2385 

15 

1500 

1611 

1722 

1833 

1944 

2389 

2500 

2611 

16 

1659 

1778 

1896 

2015 

2133 

2607 

2726 

2844 

17 

1826 

1952 

2078 

2204 

2330 

2833 

2959 

3085 

18 

2000 

2133 

2267 

2400 

2533 

3067 

3200 

3333 

19 

2181 

2322 

2463 

2604 

2744 

3307 

3448 

3589 

20 

2370 

2519 

2667 

2815 

2963 

3556 

3704 

3852 

21 

2567 

2722 

2878 

3033 

3189 

3811 

3967 

4122 

22 

2770 

2933 

3096 

3259 

3422 

4074 

4237 

4444 

23 

2981 

3152 

3322 

3493 

3663 

4344 

4515 

4685 

24 

3200 

3378 

3556 

3733 

3911 

4622 

4800 

4978 

25 

3426 

8611 

3796 

3981 

4167 

4907 

5093 

5278 

26 

3659 

3852 

4044 

4237 

4430 

5200 

5393 

5585 

27 

3900 

4100 

4300 

4500 

4700 

5500 

5700 

5900 

28 

4148 

4356 

4563 

4770 

4978 

5807 

6015 

6222 

29 

4404 

4619 

4833 

5048 

5263 

6122 

6337 

6552 

80 

4667 

4889 

5111 

5333 

5556 

6444 

6667 

6889 

31 

4937 

5167 

5396 

5626 

5856 

6774 

7004 

7233 

32 

5215 

5452 

5689 

5926 

6163 

7111 

7348 

7585 

33 

5500 

5744 

5989 

6233 

6478 

7456 

7700 

7944 

34 

5793 

6044 

6296 

6548 

6800 

7807 

8059 

8311 

35 

6093 

6352 

6611 

6870 

7130 

8167 

8426 

8685 

36 

6400 

6667 

6933 

7200 

7467 

8533 

8800 

90G7 

37 

6715 

6989 

7263 

7537 

7811 

8907 

9181 

9456 

38 

7037 

7319 

7600 

7881 

8163 

9289 

9570 

9852 

39 

7367 

7656 

7944 

8233 

8522 

9678 

9967 

10256 

40 

7704 

8000 

8296 

8593 

8889 

10074 

10370 

10667 

41 

8048 

8352 

8656 

8959 

9263 

10478 

10781 

11085 

42 

8400 

8711 

9022 

9333 

9644 

10889 

11200 

11511 

43 

8759 

9078 

9396 

9715 

10033 

11307 

11626 

11944 

44 

9126 

9452 

9778 

10104 

10430 

11733 

12059 

12385 

45 

9500 

9833 

10167 

10500 

10833 

12167 

12500 

12833 

46 

9881 

10222 

10563 

10904 

11244 

12607 

12948 

13289 

47 

10270 

10619 

10967 

11315 

11663 

13C56 

13404 

13752 

48 

10667 

11022 

11378 

11733 

12089 

13511 

13867 

14222 

49 

11070 

11433 

11796 

12159 

12522 

13974 

14337 

14700 

50 

11481 

11852 

12222 

12593 

12963 

14444 

14815 

15185 

51 

11900 

12278 

12656 

13033 

13411 

14922 

15300 

15678 

52 

12326 

12711 

13096 

13481 

13867 

15407 

15793 

16178 

53 

12759 

13152 

13544 

13937 

14330 

15900 

16293 

16685 

54 

13200 

13600 

14000 

14400 

14800 

16400 

16800 

17200 

55 

13648 

14056 

14463 

14870 

15278 

16907 

17315 

17722 

56 

14104 

14519 

14933 

15348 

15763 

17422 

17837 

18252 

67 

14567 

14989 

15411 

15833 

16256 

17944 

18367 

18789 

58 

15037 

15467 

15896 

16326 

16756 

18474 

18904 

19333 

59 

15515 

15952 

16389 

16826 

17263 

19011 

19448 

19885 

60 

16000 

16444 

16889 

17333 

17778 

19556 

20000 

20444 

TABLE  V. — CUBIC  YARDS   PER   100  FEET.     SLOPES   1%   :  1. 


303 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
.  18 

Base 
20 

Base 
28 

Base 
30 

Base 
32 

I 

50 

57 

65 

72 

80 

109 

117 

124 

2 

111 

126 

141 

156 

170 

230 

244 

259 

3 

183 

206 

228 

250 

272 

361 

383 

406 

4 

267 

296 

326 

356 

385 

504 

533 

563 

5 

361 

398 

435 

472 

509 

657 

694 

731 

6 

467 

511 

556 

600 

644 

822 

867 

911 

r* 

583 

635 

687 

739 

791 

998 

1050 

1102 

g 

711 

770 

830 

889 

948 

1185 

1244 

1304 

9 

C50 

917 

983 

1050 

1116 

138t 

1450 

1517 

10 

1000 

1074 

1148 

1222 

1296 

1593 

1667 

1741 

11 

1161 

1243 

1324 

1406 

1487 

1813 

1894 

1976 

12 

1333 

1422 

1511 

1600 

1689 

2044 

2133 

2222 

13 

1517 

1613 

1709 

1806 

1902 

2287 

2383 

2480 

14 

1711 

1815 

1919 

2022 

2126 

2541 

2644 

8748 

15 

1917 

2028 

2139 

2250 

2361 

2806 

2917 

3028 

16 

2133 

2252 

2370 

2489 

2607 

3081 

3200 

3319 

17 

2361 

2487 

2613 

2739 

2865 

3369 

3494 

3620 

18 

2600 

2733 

2867 

3000 

3133 

3667 

3800 

3933 

19 

2850 

2991 

3131 

3272 

3413 

3976 

4117 

4257 

20 

3111 

3259 

3407 

3556 

3704 

4296 

4444 

4592 

21 

3383 

3539 

3694 

3850 

4005 

4628 

4783 

4939 

22 

3667 

3830 

3993 

4156 

4318 

4970 

5133 

5296 

23 

3961 

4131 

4302 

4472 

4642 

5324 

5494 

5665 

24 

4267 

4444 

4622 

4800 

4978 

5689 

5867 

6044 

25 

4583 

4769 

4954 

5139 

5324 

6065 

6250 

6435 

26 

4911 

5104 

5296 

5489 

5681 

6452 

6644 

6837 

27 

5250 

5450 

5650 

5850 

6050 

6850 

7050 

7250 

28 

5600 

5807 

6015 

6222 

6430 

7259 

7467 

7674 

29 

5961 

6176 

6391 

6606 

6820 

7680 

7894 

8109 

30 

6333 

6556 

6778 

7000 

7222 

8111 

8333 

8555 

31 

6717 

6946 

7176 

7406 

7635 

8554 

8783 

9013 

32 

7111 

7348 

7585 

7822 

8059 

9007 

9244 

9482 

33 

7517 

7761 

8006 

8250 

8494 

9472 

9717 

9962 

34 

7933 

8185 

8437 

86S9 

8941 

9948 

10200 

10452 

35 

8361 

8620 

8880 

9139 

9398 

10435 

10694 

10954 

36 

8800 

9067 

9333 

9600 

9867 

10933 

11200 

11467 

37 

9250 

9524 

9798 

10072 

10346 

11443 

11717 

11991 

38 

9711 

9993 

10274 

10556 

10837 

11963 

12244 

12526 

39 

10183 

10472   10761 

11050 

11339 

12494 

12783 

13072 

40 

10667 

10963 

11259 

11556 

11852 

13037 

13333 

13630 

41 

11161 

11465 

11769 

12072 

12376 

13591 

13894 

14198 

42 

11667 

11978 

12289 

12600 

12911 

14156 

14467 

14778 

43 

12183 

12502 

12820 

13139 

13457 

14731 

15050 

15369 

44 

12711 

13037 

13363 

13689 

14015 

15319 

15644 

15970 

45 

13250 

13583 

13917 

14250 

14583 

15917 

16250 

16583 

46 

13800 

14141 

14481 

14822 

15163 

16526 

16867 

17207 

47 

14361 

14709 

15057 

15406 

15754 

17146 

17494 

17843 

48 

14933 

15289 

15644 

16000 

16356 

17778 

18133 

18489 

49 

15517 

15880 

16243 

16606 

16968 

18420 

18783 

19146 

50 

16111 

16481 

16852 

17223 

17592 

19074 

19444 

19815 

51 

16717 

17094 

17472 

17850 

18228 

19739 

20117 

20494 

52 

17333 

17719 

18104 

18489 

18874 

20415 

20800 

21185 

53 

17961 

18354 

18746 

19139 

19531 

21102 

21494 

21887 

54 

18600 

19000 

19400 

19800 

20200 

21800 

22200 

22600 

56 

19250 

19657 

20065 

20472 

20880 

22509 

22917 

23324 

56 

19911 

20326 

20741 

21156 

21570 

23230 

23644 

24059 

57 

20583 

21006 

21428 

21850 

22272 

23961 

24383 

24805 

58 

21267 

21696 

22126 

22556 

22985 

24704 

25133 

25563 

59 

21961 

22398 

22835 

23272 

23709 

25457 

25894 

26332 

60 

82667 

23111 

23556 

24000 

24444 

26222 

26667 

27111 

TABLE    V. CUBIC    YARDS    PICK     1<MI    KEKT.      SLOPES    2 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 

20 

Ba-e 
28 

Base 
30 

Base 
32 

1 

52 

59 

67 

74 

81 

111 

119 

126 

2 

119 

133 

143 

1C3 

178 

237 

252 

267 

3 

200 

222 

244 

267 

289 

378 

400 

422 

4 

296 

326 

356 

385 

415 

533 

563 

593 

5 

407 

444 

481 

519 

556 

704 

741 

778 

6 

533 

578 

C22 

637 

711 

889 

933 

978 

7 

674 

726 

778 

830 

881 

1089 

1141 

1193 

8 

830 

889 

94;! 

1007 

1067 

1304 

1303 

1422 

9 

1000 

1067 

1133 

1200 

1267 

1533 

1600 

1667 

10 

1185 

1259 

1333 

1407 

1481 

1778 

1852 

1926 

11 

1385 

1467 

1548 

1630 

1711 

2037 

2119 

2200 

12 

1600 

1689 

1778 

1867 

1956 

2311 

2400 

2489 

13 

1830 

1926 

2022 

2119 

2215    2600 

2696 

2793 

14 

2074 

2178 

2281 

2385 

2489    2904 

3007 

8111 

15 

2333 

5114 

2356 

2667 

2778    3222 

3'?33 

3444 

i 

2607 

2726 

2844 

2933 

3081  i  3550 

3674 

8793 

17 

2896 

3022 

3148 

3274 

3400 

3!KM 

4030 

4156 

18 

8200 

6333 

3437 

3600 

3733 

4267 

4400 

4533 

16 

8319 

3«59 

3800 

3941 

4081 

4644 

4785 

4926 

90 

3852 

4000 

4148 

4296 

4444 

5037 

5185 

5333 

21 

4200 

4356 

4511 

4667 

4822 

5444 

5600 

5756 

22 

4303 

4<V30 

4889 

5052 

5215 

5867 

6030 

6193 

23 

4941 

5111 

5281 

5452 

5622 

6304 

6474 

6644 

24 

5333 

5511 

5639 

5867 

6014 

6756 

6933 

7111  ' 

25 

5741 

5926 

6111 

6296 

6481 

7-222 

7407 

7593 

26 

6163 

6356 

6548 

6741 

6933 

7704 

7896 

8089 

27 

6600 

6800 

7000 

7200 

7400    8200 

8400 

8600 

28 

7052 

7259 

7467 

7074 

78S1 

8711 

8919 

9126 

29 

7519 

T7..3 

7948 

8163 

8378 

9237 

9452 

9667 

30 

8000 

8222 

8444 

8667 

8889 

9778 

10000 

10222 

31 

8496 

8726 

8956 

9185 

9415 

10333 

10563 

10793 

32 

9007 

9244 

9481 

9719 

9936 

10904 

11141 

11378 

33 

9538 

9r78 

10022 

10267 

10511 

11489 

11733 

11978 

34 

10074 

10326 

10578 

10330 

11081 

12089 

12341 

12593 

35 

10330 

10889 

11148 

11407 

11667 

12704 

12963 

13223 

36 

11200 

11467 

11733   12000 

12267 

13333 

13600 

13867 

87 

11785 

12059 

12333   12607 

12881 

13!»78 

14252 

14526 

38 

12385 

126H7 

12948 

13230 

13511 

14037 

14919 

15200 

39 

1300G 

13289 

13378 

13867 

14156 

15311 

15000 

1H889 

40 

13630 

13926 

14222 

14519 

14815 

16000 

16296 

16593 

41 

14274 

14578 

14881 

15185 

15489 

16704 

17007 

17311 

42 

14.133 

15244 

155:6 

15867 

16178 

17^22 

17733 

18044 

43 

15607 

15926 

1(3224 

16563 

16881 

18156 

18474 

18793 

44 

16296 

16(322 

16948 

17274 

17600 

18904 

19230 

19556 

45 

17000 

17333 

17067 

18000 

18333 

19667 

20000 

20333 

46 

17719 

18059 

18400 

18741 

19081 

20444 

20785 

21126 

47 

18452 

18SUO 

19148 

19496 

19844 

21237 

21585 

21933 

48 

19200 

19556 

19911 

20267 

20022 

22044 

22400 

22756 

49 

19963 

20326 

20689 

21052 

21415 

22867 

23230 

23593 

50 

20741 

20711 

21481 

21852 

22222 

23704 

24074 

24444 

51 

21:33 

21911 

23289 

22667 

23044 

24556 

24933 

25311 

52 

22-341 

22726 

23111 

23496 

23881 

25422 

25807 

26193 

M 

23163 

2J556 

21948 

24341 

24733 

26:304 

26696 

2r089 

54 

24000 

V4400 

24800 

25200 

2->600 

27200 

27600 

28000 

55 

24852 

25-259 

25667 

2(5074 

26481 

28111 

28519 

28926 

56 

25719 

26133 

26548 

26963 

27378 

29037 

29452 

29867 

57 

23600 

27^22 

27444 

27867 

28289 

23978 

30400 

30822- 

58 

27496 

r.^26 

28356 

23785 

29215 

30933 

31363 

31793 

59 

28407 

^8844 

29281 

29719 

30156 

31904 

32341 

32778 

60 

29333 

29778 

30222 

30667 

31111 

83889 

38333 

33778 

TAIM.I-:    V. CUI'.IC    VARHS    I'KU     1 0( )    FKKT.      SLOl'F.S    •'>:!. 


Depth 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

12 

14 

16 

18 

20    28 

30  I  32 

1 

56 

63 

70 

78 

85     115 

122 

130 

1:33 

148 

163 

178 

193 

252 

267 

281 

8 

233 

9M 

278 

300 

322 

4lf 

433     456 

4 

356 

385 

415 

444 

474 

693 

622 

652 

5 

500 

537 

574 

611 

648 

796 

888 

870 

6 

667 

711 

756 

800 

844 

1022 

1067 

1111 

7 

856 

907 

959 

1011 

1'HB 

1270 

1322 

1374 

8 

1067 

1126 

1185 

1244 

1304 

1541 

1600    1659 

9 

1300 

1367 

1433 

1500 

1567 

1833 

1900    1967 

10 

1556 

1630 

1704 

1778 

1852 

2148 

2222 

2296 

n 

18.33 

1915 

1996 

2078 

2159 

2485 

2567 

2648 

12 

2133 

2°22 

2311 

2100 

2489 

2844 

2933 

3022 

13 

2456 

2552 

2648 

2744 

2811 

3226 

3322 

3419 

14 

2800 

2904 

3007 

3111 

3215 

3630 

3733 

3837 

15 

3167 

8378 

3389 

3500 

3611 

4056 

4167 

4278 

16 

.3556 

3674 

3793 

3911 

4030 

4504 

4622 

4741 

17 

3967 

4093 

4219 

4344 

4470 

4974 

5100 

5226 

18 

4400 

4533 

4667 

4800 

4933 

5467 

5600 

5733 

19 

4856 

4996 

5137 

5278 

5419 

5981 

6122 

6263 

20 

5333 

5481 

5630 

5778 

5926 

6519 

6667 

6815 

21 

5833 

5989 

6144 

6300 

6456 

7078 

7233 

7389 

22 

6356 

6519 

6681 

6844 

7007 

7659 

7822 

7985 

23 

6900 

707'0 

7241 

7411 

7581 

8263 

8433 

8504 

24 

7467 

7644 

7822 

8000 

8178 

8889 

9067 

9144 

25 

8056 

8241 

8426 

8611 

8796 

9537 

9722 

9807 

26 

8667 

8859 

9052 

9244 

9437 

10207 

10400 

10593 

27 

9300 

9500 

9700 

9900 

10100 

10900 

11100 

11300 

28 

9956 

10163 

10370 

10578 

10785 

11615 

11822 

12030 

29 

10633 

10H48 

11063 

11278 

11493 

12352 

12567 

12781 

30 

11333 

11556 

11778 

12000 

12222 

13111 

13333 

13556 

31 

12056 

12285 

12515 

12744 

12974 

13893 

14122 

14352 

32 

12800 

13037 

13274 

13511 

13748 

14696 

14933 

15170 

33 

13567 

13811 

14056 

14300 

14514 

15522 

15767 

16011 

34 

14356 

14607 

14859 

15111 

15363 

16370 

16622 

16874 

35 

15167 

15426 

15685 

15944 

16204 

17241 

17500 

17759 

36 

16000 

16267 

16533 

16800 

17067 

18188 

18400 

18667 

37 

16856 

17130 

.17404 

.17678 

17952 

19048 

19322 

19596 

38 

17733 

18015 

18296 

18578 

18859 

19985 

20267 

20548 

39 

18633 

18922 

19211 

19500 

19789 

20944 

212&3 

21522 

40 

19556 

19852 

20148 

20444 

20741 

21926 

22222 

22516 

41 

20500 

20804 

21107 

21411 

21715 

22930 

23233 

23537 

42 

21467 

21778 

22089 

22400 

22711 

23956 

24267 

24578 

43    23456 

22774 

23093 

23411 

23730 

25004 

25322 

25641 

44 

2S467 

23793 

24119 

24444 

24770 

26074 

26400 

26726 

45 

24500 

24833 

25167 

25500 

35888 

27167 

27500 

27833 

46 

25556 

25896 

26237 

26578 

26919 

28281 

28622   28963 

47 

26038 

26981 

27:330 

27678 

28026 

29419   29767 

30115 

48 

27733  ,  28089 

28444 

28800 

29156   30578   30933 

31289 

49 

28856   89219 

28581 

29944 

30307  !  31759 

32122 

32485 

50 

30000 

30370 

30741 

31111 

31481 

32963* 

33333 

33704 

51 

31167 

31544 

31922 

32300 

32678 

34189 

34567 

34944 

52 

32356 

32741 

33126 

33511 

33896 

35437 

35822 

36207 

53 

33567 

3-3959 

34352 

34744 

35137 

36707 

37100 

87493 

54 

34800 

35200 

35600 

36000 

36400 

38000 

38400   38800 

55 

36056   31)463 

36870  ,  37278 

37685 

39315 

39722   40180 

56 

37333 

37748 

38163 

38578   38993 

40652 

41067   41481 

57 

38633 

39056 

39478 

39900 

40322 

42011 

42433   42856 

58 

39956   40385 

40815 

41244 

41674 

4a393 

43822   44252 

59 

41300   41737 

42174 

42611 

43048 

44796 

45233   45670 

60 

42667   43111 

43556 

44000 

44444 

46222   46667  j  47111 

INDEX. 


Acre  in  square  chains 18 

in  square  feet 18 

in  square  meters 18 

in  square  poles 18 

in    square    varas 18 

in   square  yards 18 

Additions,    city 173 

Adjustments,   axis  of  revolu- 
tion    36 

bubble    tube 106 

compass    36 

compass  needle 36 

compass  pivot 37 

compass  plate   bubble 36 

cross-wire   106 

plane  of  sights 37 

transit  line  of  sights 46 

transit  plate  levels 46 

wyes    107 

Agonic  line 29 

Alidade   38 

Angles   by  repetition 45 

Angular  convergence 196 

Application  of  57.3  rule 20 

Approximate   traversing 76 

Approximations   in   stadia 124 

Area,  by  coordinates 75 

of    farm 67 

of  triangle 19 

table    68 

Attachment,    compass ,   40 

Attraction,  local 35 

Average  end  areas 157 

formula  158 

Azimuth   25 

by  sun    54 

formula 53 

Back    sights 102 

Balancing  a   survey 64 

rule,    no    latitude 72 

Bearing    . 25 

and  length  lost ; 85 

how  read 27 

lost  89 

magnetic    33 

of  line 25 

true    25 

Bench  marks 104 

Horm    148 

Bibliography,   city  surveying.  189 

compass   surveying 37 

division,  of  land 94 

earthwork    168 

lettering     195 

topography    127 


Blocks,    city 180 

irregular 182 

Borrow     pits 148,   164 

Boundaries,    irregular 78 

Breaking   chair 8 

Bubble    tube  adjustment 106 

radius    107 

Cabinets  for  drawings 188 

Cases  for  city  drawings 182 

Chain,    breaking 8 

engineer's    ;  . . .     2 

Gunter's     1 

problems    10 

surveying    .' . . .     9 

vara   '• 

Chaining    7 

over   hills 9 

over  valleys 9 

Chainman,  head 8 

rear    8 

Chainmen 7 

Changes  in  declination 31 

Chart,  isogonic 30 

Circular  curve   cross-section. 185 

Circular    curves,    vertical 112 

Circumpolar   stars    51 

City   additions    173 

blocks     180 

contracts     189 

datum 178 

engineer     171 

engineer's   notes    183 

engineer's     records     185 

field    notes    186 

orders     189 

surveying    171 

Colby's  slide  rule   123 

Compass     25 

adjustments     36 

attachment     39 

bibliography     37 

tripod     26 

use    of    27 

vernier     30 

Concrete    monuments     174 

Convergence,    angular    .196 

linear    . . .« 197 

of  meridians    196 

Correction        for        erroneous 

areas     16 

lengths     16 

plot     194 

pull     13 

sag     1  f» 

1        temperature     13 


307 


308 


1XDEX. 


Courses  of  no  departure 72 

no  latitude  72 

Cox's  stadia  computer 122 

Cross-section,  streets 184 

railroad  152 

Cross-wire  adjustment 106 

Culmination,  lower  52 

upper  52 

Curvature  of  earth 109 

refracted  rav  109 

Curve  at  sag Ill 

summit  110 

Curves,  parabolic  110 

vertical  110 

Cuts,  side-hill  1 57 

Data    for   city    map 179 

on    land     plots 190 

Datum    for    city    1  <8 

piano 103 

Declination    changes    31 

for    farm     34 

how  set   off   31 

on   vernier    29 

Degree    formula     129 

of    curve    129 

Departure,     definition 61 

how    found     61 

Detail  maps  for  city 1 .187 

Diagonal   prism    56 

Dipper,    the    51 

Discrepancies    in    survey....   79 

Dividing    land     90 

Division    of    quadrilateral 83 

of    township     203 

of    triangle     81 

Double    meridian    distance...   66 
Drawing    cabinets     188 

Earthwork , 151 

bibliography     1 68 

examples    ' 159 

note    book     : . . .  162 

special 164 

Effect    df    refraction 109 

Elasticity,    modulus   of 14 

Elevation     103 

Elongation,    cast    52 

west     52 

End   areas    157 

End  of  fill 166 

Engineer's    chain    2 

Erroneous  areas 16 

lengths    15 

Error,  of  closure 63 

External    130 

Eye-piece  of  telescope 98 

Feet  to  varas 18 

Field  notes  for  city 186 

for  farm 36 

for   U.    S.    survey 205 


Foot   curves 135 

Foresights   102 

Formula  for  area  of  triangle.   19 

for    azimuth 53 

for  length  correction 16 

for  oblique  triangle 209 

1'or    offsets     198 

prismoidal     15.! 

Formulas,  approximate 133 

for   right   triangle 208 

Freehaul    167 

General  formulas,   railroad. .  .130 

maps  for  city   179 

solution    for    division 84 

Government     surveying 196 

Grade    163 

point    164 

Great    Bear    52 

Gunter's  chain    1 

Gunter's   chain   to  varas 18 

Hand-level    142 

topography  126,   143 

Height  of  instrument    102 

Hook's    law    14 

Hubs 141 

Inclined  sights    117 

Intersections    23 

Irregular    boundaries 78 

blocks    ...- 182 

section    154 

Isogonic  chart  30 

.Jacob's  staff    27 

Labor    18 

Land  plots   190 

Latitude,     definition 61 

how    found 61 

Laying-  out  curve 130 

League,    Spanish 18 

Length   of   two    courses 86 

of    curve 131 

Lenses    of    eye-piece 98 

Lettering     194 

books   195 

Level   note   books 105 

sections    153 

the  wye   96 

Leveling,   theory  of   102 

Linear  convergence   197 

units    17 

Local    attraction 35 

Location    field   book 138 

by    off-sets 132 

of  houses 22 

of    meridian 48 

of  meridian  by  Polaris 52 

1       survey 136 


INDEX. 


309 


Lost  parts    85 

Lots,    rectangular .181 

Lower  motion  of  transit 42 

Magnetic  bearing   33 

needle 25 

Maps,  detail  for  city 187 

for  city    179 

Meanders    78 

Meridian   by  Polaris    48 

by  sun    54 

distance   66 

reference   200 

without    calculation.; 57 

Metallic  Tapes    5 

Meters  to  varas   18 

Methods    of   plotting 190 

Metric  c-irves    135 

Middle   ordinate 133 

Modulus  of  elasticity 14 

A'fonuments,     kinds .173 

for    city    surveying 172 

location  of    175 

necessity    for    172 

Motion,  lower   42 

upper 42 

Napier's  laws    210 

Needle,   magnetic    25 

New  York  rod 100 

Note-book  for   earthwork. ..  .162 

for    level    105 

for  transit   137,  139 

Object   glass    98 

Objective    of    telescope 97 

Objects  of  city  survey.  ...;.  .171 

Oblique  trianglo  208 

spherical    triangle 210 

Obstacles     132 

Off-sets,    examples    200 

in   government   surveying.  .198 

intermediate     ' 199 

location   by 132 

Old  lines,  how  run 33 

Outs     IS 

Overhaul    166 

Pacing  survev 21 

Parallels,  how  run   198 

of  latitude    196 

standard    200 

Parts  of  compass 25 

of  level 96 

of  transit    38 

Peg  adjustment  of  transit...   48 

Philadelphia    rod 100 

Pins,   surveying   5 

Pit,   borrow    164 

Pivot    adjustment 37 

Plot   correction 194 

of   fsirni    .  -.193 


Plots      190 

and    lettering    190 

Plotting    by    co-ordinates.  ..  .192 
by     latitudes     and     depar- 
tures     191 

by    sines    191 

by  tangents   191 

Plumb-bob    6 

Point  of  curvature  (P.C.) 130 

of  intersection   (P.  I.) 130 

of   tangency    (P.T.)    136 

Polaris     51 

Poles  to  varas   18 

range 6 

Preliminary   earthwork    esti- 
mates     160 

note-book    136 

survey     136 

Prescriptive  rights   184 

Primary    triangulation    120 

Principal    focus    115 

focal    distance    115 

Prismoid     151 

Prismoidal  formula    151 

Private   notes   183 

Profile     ; 106 

Protractor 188 

Protractor  plotting   190 

Pull  on  tape 13 

Radius  of  bubble  tube 101 

of   curve 129 

of  parallels   196 

of    street    cross-section. . . .185 

Railroad  curve    129 

excavation     152 

surveying    129 

Range    lines     23 

poles .     6 

Ranges    202 

Reading    bearings 27 

compass  vernier 31 

of    transit    vernier 30 

rod   vernier    29 

Records,  city  engineers 185 

Rectangular  blocks    180 

lots     181 

off-sets  23 

Reduction  method   122 

tables     134 

Reference   lines    44 

meridians    200 

Refraction     55 

effect   of    109 

Reinhr.rdt's  lettering   195 

Repeating  method  for  angles  45 
Result  of  declination  changes  32 

Reticule     41 

Right  angle  by  chain 10 

plane  triangle  208 

spherical   triangle 210 


INDEX. 


Rights,   prescriptive     184 

Rod,   New  York ..100 

Philadelphia   100 

self -reading    1-02 

stadia    118 

Kuie    tor   balancing    65 

for  borrow   pits    165 

for  D.M.D 66 

for  earthwork   155 

for  finding  area 67 

for   setting  slope   stakes...  145 

of    57.3     20 

Running  parallels 198 

Sag  correction 15 

Secant  method   199 

Sections,   irregular   154 

level     153 

three-level    154 

two-level     153 

Self-reading   rod    ......102 

Setting    declination    31 

up    level    100 

up  transit   41 

Shifting   center    41 

Shrinkage   in    earthwork 167 

Side-hill    cut 157 

Slide     rule.     Colby's 123 

Slope-stakes  in  cut 144 

in  fill  145 

on   level    145 

Solar  attachment 55 

Spanish   labor  18 

league     18 

Special  case  of  earthwork. .  .164 

Square  chains  in  acre 18 

feet  in  acre   18 

poles  in  acre  18 

varas  in  acre   18 

yards   in  acre 18 

Stadia  computer   123 

formulas    115 

rod     118 

stations    120 

Stakes    for    railroad    survey.  141 

Standard    parallels    200 

Standardized   tapes    4 

Steel  tapes  3 

Stone  monuments 174 

Street  cross-section 184 

Survey  by  pacing 21 

discrepancies   79 

of  farm  by  pace  23 

topographic    114 

Surveying  by  transit  43 

city    171 

Surveyor's   compass 25 

pins    5 

Table  for  area 68 

for   level    sections 161 


traverse 
Tangent  method 


62 
199 


of  plotting    ................  191 


Tapes  for  city  surveying   .    ..175 

metallic    5 

standardized    4 

steel     3 

Telescope    101 

Temperature  correction 73 

The    57.3   rule 19 

Theory  of  leveling   1  02 

Three-level  sections    ir>4 

Tiers    202 

Topographic  field  work   119 

survey     114 

Topography  by  hand  level   ..126 

by   stadia    115 

Township   division    203 

Townships    202 

Transit     38 

as   compass    43 

essential  parts    38 

for   city   use    176 

party     .140 

plate    levels   adjusted 4fi 

surveying     43 

topography     114 

vernier    43 

Traverse  tables    62 

Traversing    7(5 

approximate     7H 

Trees,   fore  and  aft   .°,f> 

line        3f: 

witness     35 

Triangle  area   of 19 

oblique     209 

PZS    : 52 

Trigonometric  formulas   208 

Tripod,    compass    26 

Two-level   sections 153 

Unit    pull    13 

stress   14 

stretch    13 

Units  of  land  measure   18 

Upper  motion  of  transit 42 

Use  of  compass 27 

of  transit   42 

Vara    18 

Vara  chain 3 

Vernier,  compass 30 

rod    28 

transit    43 

Vertical  circle 40 

circular   curves    112 

curves    110 

Wire  interval   116 

Witness  trees   35 

Witnessing  a  corner 35 

Wye  adjustment   107 

level  .' 96 

Yard    17 

Yards  to  varas    18 

96 


I  UNIVERSITY) 


THIS   BOOK  O  ^ 

°     " 


XB  II 059 


179777 


